arXiv:1404.5871v3 [] 18 May 2016

F -threshold functions: syzygy gap fractals and
the two-variable homogeneous case
Daniel J. Hernández* and Pedro Teixeira**
* Department

of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
E-mail address: dhernan@math.utah.edu
** Department of Mathematics, Knox College, Galesburg, IL 61401, USA
E-mail address: pteixeir@knox.edu

Abstract
In this article we study F -pure thresholds (and, more generally, F thresholds) of homogeneous polynomials in two variables over a field
of characteristic p > 0. Passing to a field extension, we factor such a
polynomial into a product of powers of pairwise prime linear forms, and
to this collection of linear forms we associate a special type of function
called a syzygy gap fractal. We use this syzygy gap fractal to study, at
once, the collection of all F -pure thresholds of all polynomials constructed
with the same fixed linear forms. This allows us to describe the structure
of the denominator of such an F -pure threshold, showing in particular
that whenever the F -pure threshold differs from its expected value its
denominator is a multiple of p. This answers a question of Schwede in
the two-variable homogeneous case. In addition, our methods give an
algorithm to compute F -pure thresholds of homogenous polynomials in
two variables.

1

Introduction

Fix an arbitrary field k of characteristic p > 0, and consider a polynomial
g in k[x1 , . . . , xr ] with g(0) = 0. By utilizing properties of the Frobenius
endomorphism on the ambient polynomial ring, one may show that


a
pe
pe
a
fpt(g) := inf e : g ∈ hx1 , . . . , xr i
p
is a well-defined nonzero real number contained in the unit interval. This invariant, called the F -pure threshold of g (at the origin), was originally introduced
in [TW04], though the definition we give here follows [MTW05]. Though it is
not obvious from this definition, it turns out that the F -pure threshold of a
polynomial is always a rational number [BMS08, Corollary 2.30, Theorem 3.1].

1

Note that if, in the description of fpt(g) given above, one replaces the
e
e
e
Frobenius power m[p ] = hxp1 , . . . , xpr i of the maximal ideal m = hx1 , . . . , xr i
e
with the ordinary power mp , one would instead obtain the reciprocal of the
multiplicity of g at the origin (that is, the largest N such that g ∈ mN ). Thus,
the F -pure threshold may be thought of as a sort of “Frobenius multiplicity”,
with smaller values corresponding to “worse” singularities at the origin.
In this article we are motivated by the relationship between F -pure thresholds
and another important invariant, traditionally defined for polynomials over fields
of characteristic zero. Consider a polynomial g over a field of characteristic zero
that vanishes at the origin. By referring to a log resolution of singularities, one
may assign to g the numerical invariant lct(g), called the log canonical threshold
of g (at the origin). Like the F -pure threshold, the log canonical threshold of a
polynomial is always a nonzero rational number contained in the unit interval
and may be thought of as a measure of the singularity of g (at the origin),
with smaller values corresponding to “worse” singularities. For more on this
invariant, we refer the reader to the survey [BL04] and the references cited
therein. Throughout the rest of this article, we shall always consider polynomials
vanishing at the origin and shall omit the phrase “at the origin” when referring
to F -pure and log canonical thresholds.
Remarkably, F -pure and log canonical thresholds are intimately related.
Consider a polynomial g0 over Q and, for p  0, let gp denote the polynomial
over Fp , the field with p elements, obtained by reducing the coefficients of g0
modulo p. It follows from work of Hara and Yoshida [HY03] that fpt(gp ) ≤ lct(g0 )
and limp→∞ fpt(gp ) = lct(g0 ) (see [MTW05, Theorem 3.4]). In general, little is
known about how fpt(gp ) varies with p, and an important open conjecture predicts
that fpt(gp ) = lct(g0 ) for infinitely many primes. Motivated by understanding
the situation when fpt(gp ) 6= lct(g0 ), the following question was asked by Karl
Schwede.1
Question 1.1 (Schwede). Fix g0 ∈ Z[x1 , . . . , xr ] vanishing at the origin. Assume
fpt(gp ) 6= lct(g0 ), for a prime p  0. Write fpt(gp ) = a/b in lowest terms. Does
p divide b?
In recent work, Bhatt and Singh (and, in a subsequent generalization, NúñezBetancourt, Witt, Zhang, and the first author) have shown the following: Suppose
g0 is a polynomial over Q that is homogeneous under some N-grading and such
that the ideal generated by the partial derivatives of g0 is primary to the ideal
generated by the variables. If p  0 and fpt(gp ) 6= lct(g0 ), then the denominator
of fpt(gp ) is a not just a multiple of p but in fact a power of p [BS14, HNWZ16].
In particular, a “stronger” form of Schwede’s question has a positive answer
for such polynomials.2 As far as the authors are aware, there are no such
1 Schwede’s question was asked during the Computational Workshop on Frobenius Singularities and Invariants, held in Ann Arbor, MI, in 2012. His question, and others, can be found
at https://sites.google.com/site/computingfinvariantsworkshop/open-questions.
2 The question of under what circumstances the denominator of fpt(g ) must be a power
p
of p was asked by the first author during the aforementioned workshop.

2

general descriptions of F -pure thresholds of polynomials whenever one relaxes
the hypothesis on the ideal generated by the corresponding partial derivatives.
In this article we shed some further light on Question 1.1, answering it in
what is perhaps the simplest nontrivial case.
Theorem 1 (see Theorem 7.7). If G0 ∈ Q[x, y] is a non-constant homogeneous
polynomial,3 then Question 1.1 has a positive answer for G0 . More precisely, if
p  0 and fpt(Gp ) 6= lct(G0 ), then the minimal denominator of fpt(Gp ) is of
the form kpe , where e ≥ 1 and k divides the multiplicity of some linear factor
(over C) of G0 .
We now briefly describe the main ideas in this article. For the remainder of
this introduction, G will denote a homogeneous polynomial in k[x, y], where k is
a field of characteristic p > 0. Many of our results deal with a generalization
of F -pure thresholds called, simply, F -thresholds; this generality pays off later,
allowing us to extend our main result to polynomials that are homogeneous
under non-standard N-gradings—see Theorem 8.2. Given an ideal b ⊆ k[x, y],
the F -threshold of G with respect to b, denoted ftb (G), is a numerical invariant
describing the complexity of the hypersurface defined by G. F -thresholds generalize F -pure thresholds, in the sense that fpt(G) = fthx,yi (G) (see [MTW05]).
Rather than considering F -thresholds with respect to arbitrary ideals b, we focus
instead on the case when b is generated by two non-constant, relatively prime
forms. The motivation for this restriction is that it allows us to apply the theory
of syzygy gap fractals, introduced by Han in her thesis [Han91] and generalized
and studied by the second author in [Tei02, Tei12], and related to the theory of
p-fractals developed by Monsky and the second author [Tei02, MT04, MT06].
Over k there exists a collection of pairwise prime linear forms ` = (`1 , . . . , `n )
such that G = `a1 1 · · · `ann , for some a1 , . . . , an ∈ N>0 . In Section 4 we define a
continuous function ftb (`• ) : Rn>0 → R with the property that k 7→ ftb (`k11 · · · `knn )
whenever k = (k1 , . . . , kn ) ∈ Nn>0 . This function, called an F -threshold function
(see Definition 4.2), encodes the F -thresholds with respect to the fixed ideal b of
all homogeneous polynomials with the same linear factors as G, and will play
a key role in this article. The F -threshold function is described in terms of a
syzygy gap fractal attached to the ideal b and the linear forms `1 , . . . , `n , and
properties of syzygy gap fractals worked out in [Tei12] allow us to understand F threshold functions well enough to prove our main results. More precisely, we see
that the F -threshold function (and, hence, the F -thresholds of all homogeneous
polynomials with the same linear factors as G) is completely determined by a
family of distinguished points, called critical points. It turns out that every
coordinate of a critical point is a rational number whose denominator is a power
of p, and it is precisely this fact that allows us to say something about the
denominators of F -pure thresholds of homogeneous polynomials.
Finally, we point out that our methods are effective, and provide us with an
algorithm to compute F -pure thresholds of homogeneous polynomials in two
3 Ongoing work by the first author and Emily Witt suggests that this theorem may be false
for certain non-homogeneous polynomials in two variables.

3

variables, which has been implemented by the second author in the Macaulay2
[GS] package PosChar [BBH+ ] (see Appendix A). This implementation is remarkably efficient when the polynomial factors over a relatively small field. For
instance, if a is algebraic over F5 , satisfying a3 + a + 1 = 0, and
G = x420 y 419 (x + y)417 (x + ay)390 (x + a2 y)402 (x + a3 y)438 ∈ F5 (a)[x, y],
then our current implementation takes only about 0.3 seconds to report that
fpt(G) =

1.1

46636216675556057485911762783799675605705641779512143
.
2 · 3 · 576 · 73

Outline

This paper is organized as follows. In Section 2 we recall the basics of base p
expansions of real numbers as well as the properties of syzygy gap fractals
needed in this article. In Section 3 we introduce and study a pair of functions
∆ and Φ, the former being a special instance of a syzygy gap fractal, and in
Section 4 we use these functions to define F -threshold functions. In Section 5
we define the notion of critical points and precisely describe in which ways these
points determine the values of F -threshold functions. In Section 6 we consider
F -threshold functions attached to three linear forms. In Sections 7 and 8 we
apply our methods to the study and computations of F -thresholds of polynomials
in two variables that are homogeneous under either standard or non-standard
N-gradings.

1.2

Notations and conventions

The following are some notations and conventions used throughout:
• p denotes a prime number and q always denotes a (variable) power of p.
• k is a field of characteristic p.
• If a is an ideal of k[x, y], then a[q] denotes the qth Frobenius power of a, that
is, a[q] := hf q : f ∈ ai. Also, deg a denotes the degree or colength of a, that is,
deg a := dimk (k[x, y]/a).
• The term form is used as a synonym for nonzero homogenous polynomial. In
this context, deg H denotes the typical degree of a form H under some fixed
N-grading (usually the standard N-grading) on the ambient polynomial ring.
• If S ⊆ R, then Sq denotes the set consisting of rational elements of S with
denominator q, and Sp∞ denotes the union of all Sq .
• Vectors in Rn are denoted by bold face letters, and their components are
denoted by the same letter in regular font (e.g., u = (u1 , . . . , un )). The
canonical basis vectors of Rn are denoted by e1 , . . . , en . The vectors (0, . . . , 0)
and (1, . . . , 1) are denoted by 0 and 1.
4

• Unary operations on real numbers are extended to vectors in a componentwise
fashion (e.g., due = (du1 e, . . . , dun e)).
• Given positive integers a and d, [a % d] denotes the least positive residue of a
modulo d; i.e., [a % d] is the unique integer 1 ≤ b ≤ d such that a ≡ b mod d.

2
2.1

Background
Expansions and truncations

We review here some terminology and notation concerning real numbers.
Definition 2.1. By an expansion (base p) of a real number 0 ≤ λ ≤ 1, we mean
any expression of the form
∞
X
λe
λ=
,
pe
e=0
where the digits λe are integers between 0 and p − 1. We call such an expansion non-terminating if the sequence of digits λe is not eventually zero, and
terminating otherwise.
Remark 2.2 (Comparing expansions). Clearly, the only expansion of 0 is terminating. Moreover, a real number λ ∈ (0, 1] always has a unique non-terminating
expansion, and it also has a (unique) terminating expansion if and only if
λ ∈P
Qp∞ . In this case, the two expansions of λ are related as follows: if
r
λ = e=0 λpee is the unique terminating expansion of λ, and λr 6= 0, then
λ=

X λe
λr − 1 X p − 1
+
+
pe
pr
pe
e>r

0≤e<r

is the unique non-terminating expansion of λ.
Definition 2.3. Consider a real number λ > 0. If e ≥ 0 is an integer, we call
hλie :=

dλpe e − 1
pe

the e-th truncation of λ (base p). We adopt the convention that h0ie = 0 for
every e ≥ 0, and given a point u ∈ Rn with nonnegative coordinates, we use
huie to denote the componentwise truncation of u.
Remark 2.4 (Characterizations of truncations of positive numbers). Suppose
λ > 0. It is straightforward to verify that hλie is the unique element of Qpe with
hλie < λ ≤ hλie +

1
.
pe

This leads to an important characterization (and one we will often use without
mention): if λ > 0 and µ ∈ Qpe , then µ < λ if and only if µ ≤ hλie . Consequently,
if λ > 0, then λ = hλie + p1e if and only if λ ∈ Qpe .
5

Remark 2.5 (Truncations
in terms of expansions). Fix the unique non-terminating
P∞
expansion λ = e=1 λpee of a real number 0 < λ ≤ 1. For such an expansion, the
P
sth tail τ = e>s λpee lies in (0, 1/ps ], and therefore
&
s

dλp e =

s
X
λe
e=1

pe

!

s
X
λe

'
s

p + τp

s

=

e=1

pe

!
s

s

p + dτ p e =

s
X
λe
e=1

pe

!
ps + 1.

In other words,
hλis =

λ1
λs
+ ··· + s.
p
p

From this expression, we see that the truncations hλis form a non-decreasing
sequence that converges to λ.
Remark 2.6 (Truncations of rational numbers). Given positive integers a and d,
ape − [ape % d] [ape % d]
a e
·p =
+
,
d
d
d
where [ape % d] denotes the least positive residue of ape modulo d. Since we are
dealing with least positive residues, the second summand on the right-hand side
of the above equation lies in (0, 1], while the first summand is an integer. Thus,
substituting the above equation into Definition 2.3 shows that
DaE
d

e

=

a [ape % d]
−
.
d
dpe

Basic properties of congruences show that this expression depends only on a/d,
and not on the choice of the numerator and denominator.

2.2

Syzygy gap fractals

We gather here some definitions and results concerning syzygy gaps and syzygy
gap fractals from [Tei12] and adapt them to suit our needs. In what follows,
F, G, H ∈ R := k[x, y] are forms with no common factor—that is, there is no
non-constant polynomial in k[x, y] that divides all of F , G, and H.
Definition 2.7. Let M = R(− deg F ) ⊕ R(− deg G) ⊕ R(− deg H), so that, by
the Hilbert Syzygy Theorem, there exists an exact sequence of graded R-modules
0 → R(−m) ⊕ R(−n) → M → R → R/hF, G, Hi → 0.
The syzygy gap of F , G, and H is the nonnegative integer δ(F, G, H) := |m − n|.
Syzygy gaps are easily computed using Macaulay2 [GS] or similar software.
In Macaulay2, the following code defines a function delta that computes syzygy
gaps:

6

delta := (F ,G , H ) -> (
M := ker matrix {{ F ,G , H }};
d := degrees source generators M ;
abs ( d_0_0 - d_1_0 )
)

Perhaps one of the most important aspects of syzygy gaps is their relation
with the degrees of certain ideals, which we recall below.
SG 1 ([Tei12, Proposition 2.2], [Mon06, Lemma 1(2)]). The syzygy gap δ(F, G, H)
and the degree of the ideal hF, G, Hi are related as follows:
4 deghF, G, Hi = Q(deg F, deg G, deg H) + δ(F, G, H)2 ,
where Q(a, b, c) = 2ab + 2ac + 2bc − a2 − b2 − c2 .
The proof of SG 1 relies on the fact that the Hilbert series of R/hF, G, Hi
(and, consequently, deghF, G, Hi) can be calculated from the free resolution of
R/hF, G, Hi appearing in Definition 2.7. Though we omit the proof of SG 1, we
use a similar idea to establish the following identity.
Lemma 2.8. Let U, V ∈ k[x, y] be relatively prime forms. Then deghU, V i =
deg U deg V .
Proof. Set u = deg U and v = deg V . As U and V are relatively prime, the
sequence
0 → R(−u − v) → R(−u) ⊕ R(−v) → R → R/hU, V i → 0,
in which the second map is given by 1 7→ (V, −U ) and the third map by
(A, B) 7→ AU + BV , is exact. If Hilb(t) denotes the Hilbert series of R/hU, V i,
then the above exact sequence shows that
Hilb(t) =

1 − tu − tv + tu+v
.
(t − 1)2

Applying l’Hôpital’s rule twice, we find that Hilb(1) = uv = deg U deg V , and
the lemma follows, as deghU, V i = dimk R/hU, V i = Hilb(1).
Corollary 2.9. If F and G are relatively prime, then
δ(F, G, H)2 = 4(deghF, G, Hi − deghF, Gi) + (deg H − deg F G)2 .
Proof. Standard algebraic manipulations of SG 1 produce the identity
δ(F, G, H)2 = 4(deghF, G, Hi − deg F deg G) + (deg H − deg F G)2 ,
and the claim then follows from Lemma 2.8.
SG 2 ([Tei12, Remark 2.6], [Mon06, Lemma 2(1)]). If deg H ≥ deg F + deg G
and F and G are relatively prime, then δ(F, G, H) = deg H − deg F − deg G.
7

SG 3 ([Tei12, Proposition 2.7(2)], [Mon06, Lemma 2(2)]). If a form P ∈ k[x, y]
is prime to H, then δ(P F, P G, H) = δ(F, G, H).
SG 4 ([Tei12, Equation (2)], [Mon06, Lemma 2(3)]). If ` ∈ k[x, y] is a linear
form, then δ(F, G, H`) = δ(F, G, H) ± 1.
SG 5 ([Tei12, Proposition 2.12]). Let `1 and `2 be relatively prime linear forms,
such that F , G and H`1 `2 have no common factor. Suppose that δ(F, G, H) =
δ(F, G, H`1 `2 ) and δ(F, G, H`1 ) = δ(F, G, H`2 ). Then either δ(F, G, H) = 0 or
δ(F, G, H`1 ) = 0.
While the above results hold in arbitrary characteristic, from this point on
the assumption that k is a field of positive characteristic p will become essential.
Due to the flatness of the Frobenius map over k[x, y], we have the following:
SG 6 ([Tei12, Equation (3)]). For each q = pe we have
δ(F q , Gq , H q ) = q · δ(F, G, H).
Definition 2.10. Let `1 , . . . , `n ∈ k[x, y] be pairwise prime linear forms. A cell
(with respect to `1 , . . . , `n ) is a triple of forms C = (F, G, H) such that F , G,
and H`1 · · · `n have no common factor. If C = (F, G, 1), we shall dispense with
the third component and simply write C = (F, G).
In the remainder of this section, `1 , . . . , `n ∈ k[x, y] are fixed pairwise prime
linear forms, ` = (`1 , . . . , `n ), and C is a cell (F, G, H) with respect to the `i .
Notation. For each a = (a1 , . . . , an ) ∈ Nn , `a denotes the product `a1 1 · · · `ann .

Definition 2.11. The syzygy gap fractal δC : (Q≥0 )np∞ → Q is defined as
follows:
 
a
1
δC
= · δ(F q , Gq , H q `a ),
q
q
for each q and each a ∈ Nn . (SG 6 shows that this is well defined.)

In [Tei12] these functions were defined on [0, 1]np∞ , but it will be convenient
in this paper to extend them to (Q≥0 )np∞ .
n
Notation. The
Pntaxicab metric and norm
Pnon R are denoted by dn and k·k. That
is, d(t, u) = i=1 |ti − ui | and ktk = i=1 |ti |, for each t, u ∈ R .

SG 4 gives us the following result:

SG 7 ([Tei12, Proposition 4.2]). For each t, u ∈ (Q≥0 )np∞ we have
|δC (t) − δC (u)| ≤ d(t, u).
This shows that δC is uniformly continuous, so it extends (uniquely) to a
continuous function Rn≥0 → R. Henceforth, δC will denote this extension.
The next three results were stated in [Tei12] for the original δC , defined
on [0, 1]np∞ , but also hold for the extension to (Q≥0 )np∞ (with identical proofs,
with one exception noted below) and extend to δC : Rn≥0 → R, via density and
continuity.
8

SG 8 ([Tei12, Proposition 3.4]). For each cell (F, G, H), there exists a cell (U, V )
such that δ(F,G,H) = δ(U,V ) .
Definition 2.12. Two points of (Q≥0 )nq are adjacent if they differ by ±ei /q,
for some i, where e1 , . . . , en denote the canonical basis vectors. Equivalently,
two points of (Q≥0 )nq are adjacent if the taxicab distance between them is 1/q.
SG 9 ([Tei12, Theorem II]). Suppose the restriction of δC to (Q≥0 )nq attains a
local maximum at u0 , in the sense that the values of δC at all points of (Q≥0 )nq
adjacent to u0 are smaller than δC (u0 ). Then
δC (t) = δC (u0 ) − d(t, u0 ),
for all t ∈ Rn≥0 with d(t, u0 ) ≤ δC (u0 ). In particular, δC is piecewise linear on
that region and has a local maximum at u0 in the usual sense.
The following result was first obtained by Monsky in the case where C = (x, y)
[Mon06, Corollary 9], and subsequently generalized by the second author:
SG 10 ([Tei12, Theorem III]). Suppose δC has a local maximum at a/q, where
q > 1 and a ∈ Nn has some coordinate not divisible by p. Then
 
a
n−2
δC
≤
.
q
q
As the proof of [Tei12, Theorem III] uses the fact that the δC are defined
on a unit hypercube in an essential way, relying on symmetry and reflections,
this requires explanation.
Let a/q be as in the above statement. Set b = ba/qc

and Cb = F, G, H`b . Then δC (t) = δCb (t − b) for each t with t − b ∈ Rn≥0 ,
so δCb has a local maximum at a/q − b. Since some ai is prime to p, so is the
corresponding numerator ai − bi q of a/q − b. Thus, the hypotheses of [Tei12,
Theorem III] hold for δCb at the point a/q − b ∈ [0, 1]nq , and that result shows
that δC (a/q) = δCb (a/q − b) ≤ (n − 2)/q.

Remark 2.13. The function δC is linear outside a bounded subset of Rn≥0 . To see
that, we may assume that C = (U, V ), by SG 8, and SG 3 allows us to assume that
U and V are relatively prime; SG 2 then shows that δC (a/q) = ka/qk − deg U V
whenever ka/qk ≥ deg U V . By continuity, δC (t) = ktk − deg U V , for each
t ∈ Rn≥0 with ktk ≥ deg U V .

Definition 2.14. The trivial region of δC is the set {t ∈ Rn≥0 : ktk ≥ deg U V },
where U and V are forms such that δC = δ(U,V ) (see SG 8). The complement of
the trivial region in Rn≥0 is the nontrivial region of δC .
We close this section with some consequences of SG 9.
Proposition 2.15. Suppose t0 lies in the nontrivial region of δC and δC (t0 ) > 0.
Then δC attains a local maximum at a point u0 ∈ Qnp∞ with d(t0 , u0 ) < δC (u0 ).
In particular, δC (t) = δC (u0 ) − d(t, u0 ) on a neighborhood of t0 .
9

Proof. Using the continuity of the function t 7→ δC (t) − d(t, t0 ) and the density
of Qnp∞ in Rn , choose u in some Qnq such that δC (u) − d(u, t0 ) > 0. Then
δC (u) > d(u, t0 ) ≥ 0, and SG 7 shows that δC > 0 on the line segment joining u
and t0 . Since δC vanishes on the boundary of its trivial region (see Remark 2.13),
this shows that u is also in the nontrivial region. Consider the equivalence relation
on the set S = {v ∈ (Q≥0 )nq : δC (v) > 0} generated by adjacency—that is, the
smallest equivalence relation on S containing all adjacent pairs (v, v0 ) ∈ S 2 . We
shall show that the equivalence class of u is contained in the nontrivial region,
and is therefore finite. For that, it suffices to show that points z, w ∈ S, one
in the trivial region and the other in the nontrivial region, cannot be adjacent.
Indeed, if they were adjacent, then |δC (z) − δC (w)| = 1/q = d(z, w), by SG 4,
and SG 7 would imply that δC is linear on the line segment joining z and w.
This is impossible, since δC (z) and δC (w) are both positive, and δC vanishes on
the boundary of the trivial region.
Since the equivalence class of u is finite, we can choose a point u0 in this
class where δC is maximum. Then SG 9 shows that δC attains a local maximum
at u0 and that δC (u) = δC (u0 ) − d(u, u0 ). To complete the proof, note that
d(t0 , u0 ) ≤ d(t0 , u) + d(u, u0 ) < δC (u) + (δC (u0 ) − δC (u)) = δC (u0 ).
Remark 2.16. The above proof shows that if u ∈ Qnq lies in the nontrivial region
of δC and δC (u) > 0, then the local maximum u0 that determines the behavior
of δC near u is also in Qnq .
Two corollaries follow immediately:
Corollary 2.17. The function δC is piecewise linear with coefficients in Qp∞
on each connected component of its positive locus.
Corollary 2.18. The local maxima of δC are attained at points in Qnp∞ .

3

The functions ∆ and Φ; the upper and lower
regions

Throughout this and the next two sections, ` is an n-tuple (`1 , . . . , `n ) of pairwise
prime linear forms in k[x, y], and b is an ideal of k[x, y] generated by relatively
prime non-constant forms U and V .
Definition 3.1. We use ∆ to denote δ(U,V ) , the syzygy gap fractal associated
with the cell (U, V ) with respect to the linear forms `1 , . . . , `n . That is, ∆ is the
unique continuous function Rn≥0 → R such that
 
a
1
∆
= · δ(U q , V q , `a ),
q
q
for each q and for each a ∈ Nn . We use Φ to denote the unique continuous
function Rn≥0 → R such that
 
a
1
Φ
= 2 · deghU q , V q , `a i,
(3.1)
q
q
10

for each q and for each a ∈ Nn .
Because k[x, y] is regular of dimension 2, for each ideal a of k[x, y] we have
deg a[p] = p2 · deg a. Thus, Φ(a/q) is well defined, and (3.1) gives us a function
(Q≥0 )np∞ → Q. Below we shall justify the existence of the continuous extension
of this function to Rn≥0 , tacitly assumed above. Furthermore, we shall see that
∆ and Φ are both independent of the choice of the two homogeneous generators
of the ideal b, so they can be thought of as functions attached to ` and b.
Lemma 3.2. There exists a unique continuous extension to Rn≥0 of the function
Φ : (Q≥0 )np∞ → Q defined by (3.1). Furthermore, both ∆ and Φ depend only on
the ideal b, and not on the particular choice of generators U and V .
Proof. Corollary 2.9 shows that ∆ and Φ are related as follows on (Q≥0 )np∞ :
∆(t)2 = 4(Φ(t) − deg b) + (ktk − deg U V )2 .

(3.2)

Since ∆ is defined and continuous on Rn≥0 , (3.2) can be used to (uniquely)
extend Φ to a continuous function Rn≥0 → R, establishing the first claim. To
prove the second claim, first note that deg U V depends
only on b, and not on
√
the particular choice of generators U and V : as b = hx, yi, any two forms
generating b must be a minimal set of generators; consequently, their degrees
are univocally determined by b. Next, observe that, as the restriction of Φ
to (Q≥0 )np∞ is clearly independent of the choice of U and V , so is Φ itself, by
continuity. Finally, since both Φ and deg U V are independent of the choice of
the generators U and V , then so is ∆, by (3.2).
Example 3.3. The function Φ attached to ` = (x, y) and m = hx, yi is as
follows:
(
 
a1
+ aq2 − aq1 a2 2 if a1 , a2 < q
a
1
q q
a1 a2
= 2 · deghx , y , x y i = q
Φ
q
q
1
otherwise
for each q and each a ∈ N2 ; so
(
t1 + t2 − t1 t2
Φ(t) =
1

if t1 , t2 < 1
otherwise

for each t ∈ R2≥0 , by continuity.
Definition 3.4. For u, v ∈ Rn we write u ≤ v if ui ≤ vi , for each i. The
relations ≥, <, and > on Rn are defined likewise.
Proposition 3.5 (Basic properties of Φ).
(1) The function Φ is (weakly) increasing: t ≤ u ⇒ Φ(t) ≤ Φ(u).
(2) 0 ≤ Φ(t) ≤ deg b, for each t.
11

(3) Φ(t) = deg b ⇔ ∆(t) = |ktk − deg U V |. In particular, Φ(t) = deg b for
each t such that ktk ≥ deg U V .
Proof. Points (1) and (2) are clear for t, u ∈ (Q≥0 )np∞ , and extend to all t, u ∈
Rn≥0 by continuity. The first statement in (3) follows immediately from (3.2),
while the second statement follows from the first and Remark 2.13, which states
that ∆(t) = ktk − deg U V whenever ktk ≥ deg U V .

Definition 3.6. The set T = t ∈ Rn≥0 : ktk ≥ deg U V is the trivial region
attached to b, and its complement in Rn≥0 is the nontrivial region.
Convention. All topological notions used will refer to the subspace topology of
Rn≥0 induced by the standard topology of Rn .
Notation. If X ⊆ Rn≥0 , then X and ∂X denote the closure and the boundary of
X in Rn≥0 .
Definition 3.7. The upper and lower regions attached to ` and b are the sets

U = t ∈ Rn≥0 : Φ(t) = deg b = Φ−1 (deg b)
and

L = t ∈ Rn≥0 : Φ(t) < deg b = Φ−1 ([0, deg b)).

The set B is the common boundary ∂U = ∂L of those regions in Rn≥0 .
Very often the choice of ` and b will be clear from the context (or fixed in
advance, as in this section), so we shall omit the phrase “attached to ` and b”
and ask that the reader rely on the context to determine the exact setup.
Remark 3.8. If a ∈ Nn and q is a power of p, then
a/q ∈ U ⇔ `a ∈ b[q]

and a/q ∈ L ⇔ `a 6∈ b[q] .

Example 3.9. In the setting of Example 3.3, the lower region L is the square
[0, 1)2 . Less trivial instances can be seen in Examples 4.7, 6.1, and 7.13.
Definition 3.10. Let u, v ∈ Rn . Then
[u, v] := {t ∈ Rn : u ≤ t ≤ v} = [u1 , v1 ] × · · · × [un , vn ].
The “intervals” (u, v), [u, v), [u, ∞), etc., are defined analogously.
Some properties of the regions U , L , and B follow immediately from
Proposition 3.5 and the continuity of Φ:
Corollary 3.11 (Basic properties of the regions U , L , and B).
(1) L is open and U and B are closed in Rn≥0 .

(2) U contains the trivial region T = t ∈ Rn≥0 : ktk ≥ deg U V .
(3) L is contained in the nontrivial region, and is therefore bounded.
12

(4) If u ∈ U , then [u, ∞) ⊆ U .
(5) If u ∈ L , then [0, u] ⊆ L .
(6) If u ∈ B, then [0, u) ⊆ L and [u, ∞) ⊆ U .
Remark 3.12. The lower region, in the case where b = hx, yi, was studied by
Pérez [Pér13] under a different guise—as the first constancy region of the mixed
test ideals τ (`t ).

4

The F -threshold function

Let ` = (`1 , . . . , `n ) and b = hU, V i ⊆ k[x, y] be as in the previous section. As
before, we use ∆ and Φ to denote the unique continuous functions Rn≥0 → R
such that ∆(a/q) = q −1 · δ(U q , V q , `a ) and Φ(a/q) = q −2 · deghU q , V q , `a i, for
every a ∈ Nn and for every q = pe .
Discussion 4.1. The F -threshold of a polynomial G ∈ m = hx, yi with respect
to b can be defined as follows:

ftb (G) = inf k/q ∈ (Q>0 )p∞ : Gk ∈ b[q] .
Though it is not at all obvious from this definition, it turns out that ftb (G) is
a rational number (see [BMS08, Corollary 2.30, Theorem 3.1]). When b = m,
ftb (G) is the F -pure threshold of G, denoted by fpt(G).
The condition Gk ∈ b[q] is equivalent to deghU q , V q , Gk i = deghU q , V q i
(where deghU q , V q i = deg b[q] = q 2 · deg b), so in the special case where G = `a ,
for some a ∈ Nn>0 , we find that


ftb (`a ) = inf kq ∈ (Q>0 )p∞ : Φ kq · a = deg b .
The continuity of Φ and the density of (Q>0 )p∞ in R>0 then allow us to write
ftb (`a ) = min{λ ∈ R>0 : Φ(λa) = deg b} = min{λ ∈ R>0 : λa ∈ U },
which motivates the following definition.
Definition 4.2. For each t ∈ Rn>0 we define


ftb (`t ) = min λ ∈ R>0 : Φ(λt) = deg b = min λ ∈ R>0 : λt ∈ U .
These descriptions agree, by definition of the upper region U , and the latter
description shows that ftb (`t ) is well defined: as U is closed and its complement
L is bounded (see Corollary 3.11), the minima appearing in these definitions
exist. The function t 7→ ftb (`t ) is the F -threshold function attached to ` and b.
The following are alternate characterizations of ftb (`t ):

ftb (`t ) = min λ ∈ R>0 : ∆(λt) = |λktk − deg U V |

= max λ ∈ R>0 : λt ∈ L
= the unique λ ∈ R>0 such that λt ∈ B.
13

The last characterization, which depends on Corollary 3.11(6), is perhaps the
best way to think of the F -threshold function: ftb (`t ) is the exact factor by
which t needs to be scaled to obtain a point of the boundary B.
UV
Remark 4.3. Fix t ∈ Rn>0 . The observation that degktk
· t has norm deg U V
(and hence lies in the trivial region T , and therefore in U ) shows that the factor
by which t needs to be scaled to obtain a point of the boundary B is at most
deg U V /ktk. Restated, we see that

ftb (`t ) ≤

deg U V
.
ktk

In future sections we shall try to understand in what ways the value of ftb (`t )
differs from this natural upper bound.
Example 4.4. In the situation of Examples 3.3 and 3.9 (m = hx, yi and
` = (x, y)), the closure of the lower region is the unit square [0, 1]2 , so



1 1
m t
2
ft (` ) = max λ ∈ R>0 : λt ∈ [0, 1] = min
,
.
t1 t2
Directly from the definition we have:
Proposition 4.5. For each λ ∈ R>0 and t ∈ Rn>0 we have
ftb (`λt ) =

1
· ftb (`t ).
λ

The above result and the rationality of F -thresholds of polynomials imply
that ftb (`t ) is rational whenever t ∈ Qn>0 .
Proposition 4.6. The F -threshold function is continuous.
Proof. To show continuity at c ∈ Rn>0 , Proposition 4.5 allows us to scale c
and assume that c ∈ B or, equivalently, ftb (`c ) = 1. If t ∈ Rn>0 , then the
factors by which t needs to be scaled to obtain points of ∂[0, c] and ∂[c, ∞) are,
respectively, min{ci /ti } and max{ci /ti }. By Corollary 3.11(6), the scaling factor
needed to obtain a point of B lies somewhere in between:




c1
cn
c1
cn
min
,...,
≤ ftb (`t ) ≤ max
,...,
.
(4.1)
t1
tn
t1
tn
(Figure 1 illustrates this for n = 2.) As t → c, both min{ci /ti } and max{ci /ti }
tend to 1, so ftb (`t ) → 1 = ftb (`c ), showing continuity at c.
Notation. Let x ∈ Rn≥0 and r ∈ R>0 . Then Br (x) and B r (x) denote the open
and closed taxicab balls (in Rn≥0 ) of radius r centered at x.

14

c1
u1 u
c2
u2 u

c2

U
c
c2
t2 t
c1
t1 t

L
u
t

c1

Figure 1: An illustration of inequalities (4.1) in the case n = 2
8

8

6

6

4

4

2

2

c

0

0

2

4

6

0

8

(a)

0

2

4

6

8

(b)

Figure 2: The syzygy gap fractal of Example 4.7 and its upper and lower regions
Example 4.7. Let b = hx3 +y 3 +xy 2 , x2 y 3 i ⊆ F5 [x, y] and ` = (x, y). Figure 2(a)
shows a density plot of the function ∆ attached to ` and b on the square
[0, 8]2 , where the color encodes the values of the function at each point—the
lighter the color, the larger the value. Figure 2(b) shows the corresponding
upper and lower regions. This is a rather simple example of a syzygy gap
fractal, that is entirely determined by its values at the points with integer
coordinates. For instance, as the plot suggests, ∆ attains a local maximum
at c = (2, 3). Indeed, direct computations using SG 2 and SG 3 show that
∆(c) = δ(x3 + y 3 + xy 2 , x2 y 3 , x2 y 3 ) = 3, while ∆ = 2 at each adjacent point
in N2 . SG 9 then confirms that ∆ has a local maximum at c, and also shows
that ∆(t) = 3 − |t1 − 2| − |t2 − 3| on B 3 (c). Similar formulas can be obtained
for each other local maximum in N2 , and all such formulas put together give an
explicit piecewise formula for ∆ on its nontrivial region. On the trivial region
T = {t ∈ R2≥0 : ktk ≥ 8}, on the other hand, we know that ∆(t) = ktk − 8.
Returning to c = (2, 3), note that on B 3 (c) we have ∆(t) = 8 − ktk if and
15

only if t ≥ c. Proposition 3.5(3) then shows that U ∩ B 3 (c) = [c, ∞) ∩ B 3 (c),
and thus B ∩ B 3 (c) = ∂[c, ∞) ∩ B 3 (c). Consequently, ftb (`t ) = max{2/t1 , 3/t2 }
on the cone over ∂[c, ∞) ∩ B 3 (c), which appears shaded in Figure 2(b).
The point c plays a special role because it is a local maximum “adjacent to
the trivial region”. There are five such points in this example: c1 = c = (2, 3),
c2 = (0, 7), c3 = (1, 6), c4 = (5, 1), and c5 = (7, 0), all shown in Figure 2(b). Each
of those points determines the F -threshold functionSlocally, and together they
5
determine the F -threshold function globally: U = i=1 [ci , ∞), and therefore
b t
ft (` ) equals the minimum value of the set
n
o





max t21 , t32 , max t01 , t72 , max t11 , t62 , max t51 , t12 , max t71 , t02 .
Special points like those in the above example will be examined in detail
in the next section. As we shall see, Example 4.7 is illustrative of the general
situation in that the lower region has a “staircase” aspect determined by such
special points, and that the F -threshold function is determined by those points.
But Example 4.7 is also misleadingly simple. In more general situations there
are typically infinitely many of those special points; very often those points do
not have integer coordinates, and U is not the union of boxes [c, ∞) determined
by them.

5

Critical points

Throughout this section we fix ` = (`1 , . . . , `n ) and b = hU, V i as in Section 4,
and again use ∆ and Φ to denote the functions Rn≥0 → R defined in terms of
these choices, as in Definition 3.1.
Definition 5.1. A point c ∈ Rn≥0 is a critical point associated with ` and b
if ∆ attains a local maximum at c and that local maximum is adjacent to the
trivial region T , in the sense that ∆(c) = deg U V − kck (so the ball B ∆(c) (c)
on which ∆ is determined by c touches the trivial region; see Figure 3).
If ` and b have been fixed (as they were in this section) or are clear from
the context, we shall call those points simply critical points. By Corollary 2.18,
all critical points lie in Qn
p∞ ; this observation will be fundamental in the
proofs of our main results, in Sections 7 and 8. Critical points forge the
(often extremely complex) boundary B locally—if c is a critical point, then B
agrees with the (very simple) boundary of [c, ∞) near c, as shown in the next
proposition. Consequently, the F -threshold function agrees with the function
t 7→ max{c1 /t1 , . . . , cn /tn } near c. (Proposition 5.3 will make precise what
“near” means.)
Proposition 5.2. Let c be a critical point, and set r = ∆(c).
(1) For t ∈ B r (c) we have ∆(t) = deg U V − ktk if and only if t ≥ c.
Consequently,
16

T

ktk ≥ deg U V

U
c

∆(c)

L
Figure 3: A critical point c
(2) U ∩ B r (c) = [c, ∞) ∩ B r (c),
(3) B ∩ B r (c) = ∂[c, ∞) ∩ B r (c) (so, in particular, c ∈ B), and
(4) [c, ∞) ⊆ U .

Proof. If t ∈ B r (c), then ∆(t) = ∆(c) − d(t, c) = deg U V − kck − kt − ck, by
SG 10 and the definition of critical point, so ∆(t) = deg U V − ktk if and only
if kck + kt − ck = ktk. As t and c have nonnegative coordinates, the latter
condition is equivalent to t ≥ c, giving (1). Each t ∈ B r (c) has norm less
than or equal to deg U V , so t ∈ U if and only if ∆(t) = deg U V − ktk, by
Proposition 3.5(3); parts (2)–(4) now follow easily from (1).
Proposition 5.3. If t ∈ Rn>0 and c is a critical point, then


c1
cn
deg U V
ftb (`t ) = max
· t.
,...,
⇐⇒ c ≤
t1
tn
ktk
Proof. Note that c ≤ (deg U V /ktk)t if and only if max{ci /ti } ≤ deg U V /ktk.
So, in what follows we shall prove the following equivalent assertion:




c1
cn
c1
cn
deg U V
b t
ft (` ) = max
,...,
⇐⇒ max
,...,
≤
.
(5.1)
t1
tn
t1
tn
ktk
Set γ = max{ci /ti }. Assuming ftb (`t ) = γ, Remark 4.3 implies that γ =
ftb (`t ) ≤ deg U V /ktk. To establish the remaining implication, suppose γ ≤
deg U V /ktk. By definition of γ, we have γt ∈ ∂[c, ∞). In particular, γt ≥ c,
and combining this with our assumption that γ ≤ deg U V /ktk we see that
kγt − ck = γktk − kck ≤ deg U V − kck = ∆(c),
where the last equality follows from the fact that c is a critical point. In summary,
we have just seen that γt ∈ ∂[c, ∞) ∩ B ∆(c) (c), and Proposition 5.2(3) then
shows that γt ∈ B as well. The fact that ftb (`t ) = γ then follows, as ftb (`t ) is
the unique factor by which t is scaled to obtain an element of B.
17

We now move towards the proof of some simple characterizations of critical
points. For that, we need the following elementary result.
Lemma 5.4. Let F, G ∈ R := k[x, y] be relatively prime forms of degrees a
a+b−1
and b, respectively. Then hx, yi
⊆ hF, Gi.
Proof. As F and G are relatively prime, the k-linear map Rb−1 ×Ra−1 → Ra+b−1 ,
(A, B) 7→ AF + BG, is injective. But the k-dimensions of the domain and the
codomain of the map are the same, so the map is surjective as well.
Remark 5.5. Lemma 5.4 shows, in particular, that (deg U V −1)ei ∈ U , for each i,
where e1 , . . . , en denote the canonical basis vectors. This and Corollary 3.11(3)
show that the lower region L and the boundary B are contained in the intersection of the cube [0, deg U V − 1]n and the half-space {t ∈ Rn : ktk ≤ deg U V }.
Proposition 5.6. Let c = a/q ∈ (Q≥0 )nq . The following are equivalent:
(1) c is a critical point;
(2) c ∈ U and c − t ei ∈ L , for each i such that ci > 0 and 0 < t ≤ ci ;
(3) c ∈ U and c − ei /q ∈ L , for each i such that ci > 0;
(4) `a ∈ b[q] and `a−ei 6∈ b[q] , for each i such that ai > 0.
Proof. Proposition 5.2(2) and Corollary 3.11(5) show that (1) ⇒ (2); (2) ⇒ (3)
is immediate; Remark 3.8 shows that (3) ⇔ (4). We now show that (3) ⇒ (1).
Rewrite equation (3.2) as follows:
(deg U V − ktk − ∆(t)) (deg U V − ktk + ∆(t)) = 4(deg b − Φ(t)) .
A

B

(5.2)

C

Substituting t = c, the first clause of (3) shows that C = 0. Moreover, as
q deg U V −1
hx, yi
⊆ b[q] , by Lemma 5.4, the second clause of (3) implies that
kck < deg U V , so B > 0. It follows that A = 0, so ∆(c) = deg U V − kck > 0.
It remains to prove that ∆ attains a local maximum at c. Note that SG 4 tells
us that each ∆(c ± ei /q) is either ∆(c) − 1/q or ∆(c) + 1/q. We claim that the
latter cannot happen; the result will then follow from SG 9.
Substituting t = c+ei /q in (5.2) and supposing that ∆(c+ei /q) = ∆(c)+1/q
we reach an impossibility: A = −2/q and B = 2∆(c) > 0, while C = 0 (since
c + ei /q ∈ U ). If ci > 0, then substituting t = c − ei /q in (5.2) and supposing
that ∆(c − ei /q) = ∆(c) + 1/q we again reach an impossibility: A = 0, while
C > 0, by the second clause of (3). This establishes the claim.
The above proposition shows that a critical point is a minimal point of
U with respect to the componentwise order ≤ introduced in Definition 3.4.
Although the converse does not hold—there are minimal points of U that are
not critical points—it is true that a minimal point of U ∩ Qnq is a critical point.
This gives us a method, however impractical, to locate critical points: start with

18

a point u ∈ U ∩ Qnq , and successively subtract 1/q from the coordinates of u,
until a minimal point of U ∩ Qnq is found. For a more sophisticated method to
locate critical points, see Appendix A.
A consequence of the minimality discussed above is that if b and c are critical
points with b ≤ c, then b = c. We shall use this fact often, without further
comment.
Corollary 5.7. Let u ∈ (Q≥0 )nq . Then u ∈ U if and only if there exists a
critical point c ∈ (Q≥0 )nq such that c ≤ u.
Proof. If u ∈ U , choose c ∈ U ∩ [0, u] ∩ Qnq of minimum norm; then c satisfies
condition (3) of Proposition 5.6, and is therefore the desired critical point. The
converse follows directly from Proposition 5.2(4).
Remark 5.8. While a point in U may be strictly greater
than several critical

points, that is never the case for points in ∂T = t ∈ Rn≥0 : ktk = deg U V .
Indeed, suppose b =
6 c are critical points with b, c ≤ u ∈ ∂T . Consider the
sum d(u, b) + d(u, c). Since u ∈ ∂T , this sum equals ∆(b) + ∆(c), which is
less than or equal to d(b, c), by SG 9. The triangle inequality then implies
that d(u, b) + d(u, c) = d(b, c). This fact, the assumption
that
Pn
Pnb, c ≤ u, and
the identity |b − c| + b + c = 2 max{b, c} show that i=1 ui = i=1 max{bi , ci }.
But ui ≥ max{bi , ci }, for each i, and thus ui = max{bi , ci }, for each i. Since
u 6= b, c, this shows, in particular, that neither b nor c is strictly smaller than u.
The facts proven above lead up to the following “structure theorem” for the
F -threshold function.

Theorem 5.9. Let ` = (`1 , . . . , `n ), where the `i are pairwise prime linear forms
in k[x, y], and b = hU, V i, where U and V are relatively prime non-constant
forms in k[x, y]. Fix t ∈ Rn>0 and set λ = deg U V /ktk. If, for some e, the
truncation hλtie lies in U , then there exists a unique critical point c ≤ hλtie
associated with ` and b with coordinates in Qpe , and ftb (`t ) is determined by c:


c1
cn
ftb (`t ) = max
,...,
< λ.
t1
tn
Otherwise, ftb (`t ) is determined by the trivial region attached to b:
ftb (`t ) = λ =

deg U V
.
ktk

Proof. If hλtie ∈ L , for each e, then λt ∈ L . As λt ∈ ∂T ⊆ U , λt ∈ B,
whence ftb (`t ) = λ. If hλtie ∈ U , for some e, then there exists a critical
point c = a/pe ≤ hλtie < λt, by Corollary 5.7, and Proposition 5.3 shows that
ftb (`t ) = max{ci /ti }.
The way the F -threshold function is determined by critical points (and the
trivial region) can be compactly stated as follows.

19

Corollary 5.10. For each t ∈ Rn>0 ,



 

c1
cn
deg U V
ftb (`t ) = min max
,...,
: c is a critical point ∪
.
t1
tn
ktk
Remark 5.11. Suppose that c is a critical point, so that, in particular, ∆(c) =
deg U V − kck. It follows from the inequality kck/ktk ≤ max{ci /ti } that


∆(c)
deg U V
kck
deg U V
c1
cn
=
−
≥
− max
,...,
.
(5.3)
ktk
ktk
ktk
ktk
t1
tn
If, in addition, we suppose that max{ci /ti } ≤ deg U V /ktk, then (5.1) and (5.3)
give us the following:
∆(c)
deg U V
− ftb (`t ) ≤
.
ktk
ktk

This will be particularly relevant to us in the case where c is not in Nn ; then we
can write c = a/q, where q > 1 and some ai is prime to p, and SG 10 gives
n−2
deg U V
− ftb (`t ) ≤
.
ktk
qktk

6

The case n = 3: the “Sierpiński staircases”

In this section we examine the case n = 3 in detail. Let `1 , `2 , `3 ∈ k[x, y] be
pairwise prime linear forms, ` = (`1 , `2 , `3 ), m = hx, yi, and b = hU, V i, where
U, V ∈ k[x, y] are relatively prime non-constant forms.
Example 6.1. Figure 4 shows the lower regions attached to ` = (`1 , `2 , `3 )
and m in various characteristics. Since m is invariant under linear changes of
variables, those regions do not depend on the choice of `; in fact, through a
change of variables we may assume that ` = (x, y, x + y). Figure 5 shows the
lower regions attached to ` = (x, y, x + y) and various ideals b in characteristic 5.
Because of the conspicuous presence of Sierpiński gaskets and analogues, we call
those regions “Sierpiński staircases”. These pictures were obtained by choosing a
reasonably large q and finding all maximal points v of L ∩ Q3q , and then plotting
all “boxes” [0, v] together, where v ranges over all those maximal points. For
more details (e.g., regarding the presence of the Sierpiński gaskets and analogues),
see Theorem 6.7 and Corollary 6.9.
Remark 6.2. SG 10 tells us that if the function ∆ attached to ` and b attains a
local maximum at a/q, where q > 1 and some ai is prime to p, then ∆(a/q) =
1/q. This is reflected in a unique feature of critical points in the case n = 3.
Suppose c = a/q, with a and q as above, is a critical point. As ∆(c) = 1/q,
kck + 1/q = deg U V , by definition of critical point. That is, c is at distance 1/q
from the trivial region.

20

Figure 4: Sierpiński staircases in characteristics 2, 3, and 5

Figure 5: Sierpiński staircases in characteristic 5 attached to ` = (x, y, x + y)
and the ideals hx3 , y 4 i, hx5 , y 7 i, and hx3 + y 3 , x2 yi
Remark 6.3 (Coordinate sums of truncations). Consider a point u ∈ R3 with
positive coordinates. As in Remark 2.4, we have that
huis < u ≤ huis +

1
ps

for every s ≥ 0, where 1 denotes the vector (1, 1, 1). The second inequality is an
equality in some component if and only if that component lies in Qps . From this,
it follows that when kuk ∈ N, khuis k = kuk − a/ps for some integer 1 ≤ a ≤ 3,
and that a = 3 if and only if u ∈ Q3ps .
With these observations, we are ready to begin our examination of the
intersection of the boundary B and the trivial region.

21

Lemma 6.4. Fix u ∈ R3>0 with kuk = deg U V and hui0 ∈ L . If huie ∈ U for
some e ≥ 1, then there exists 1 ≤ s ≤ e such that huis is a critical point with
khuis k = deg U V − 1/ps .
Proof. If huie ∈ U , then Corollary 5.7 shows that there exists a critical point
c ∈ Q3pe with c ≤ huie < u. If c were integral, then we would have that c ≤ hui0 ,
which contradicts the assumption that hui0 ∈ L . Consequently, the minimal s
such that c ∈ Q3ps must satisfy 1 ≤ s ≤ e, and so, by Remark 6.2,
kck +

1
= deg U V = kuk.
ps

Because u − c > 0 and ku − ck = 1/ps , we conclude that u − c < 1/ps , so
c < u < c + 1/ps , and c = huis , by Remark 2.4.
Theorem 6.5. Fix u ∈ R3>0 with kuk = deg U V and hui0 ∈ L , and set


1
µ = inf s ≥ 1 : khuis k = deg U V − s .
p
If µ = ∞, then u ∈ B. Otherwise, huiµ is a critical point, and u ∈
/ B.
Proof. If µ = ∞, then khuis k =
6 deg U V − 1/ps for every s ≥ 1; Lemma 6.4 then
implies that huie ∈ L for all e ≥ 1, and so u ∈ L . Since u lies also in the
trivial region, we conclude
that u ∈ B.
µ
µ
If µ < ∞, then ` p huiµ lies in mdeg U V ·p −1 , which by Lemma 5.4 is contained
µ
µ
µ
in hU p , V p i = b[p ] . In other words, huiµ ∈ U , and the minimality of µ and
Lemma 6.4 further show that huiµ must be a critical point.
Definition 6.6. The Sierpiński p-gasket Sp consists
of all points v ∈ [0, 1]3 with
P∞
4
e
kvk = 2 for which there exists an expansion v = e=1 w
pe with kwe k = 2p − 2
for all e ≥ 1.
Alternatively, let L = (0, 1, 1), M = (1, 0, 1), and N = (1, 1, 0). Any point
v ∈ [0, 1]3 with kvk = 2 can be written uniquely as v = λL + µM + νN, where
λ, µ, ν ∈ [0, 1] and λ + µ + ν = 1, and it is straightforward to verify that v ∈ Sp
if and only if λ, µ and ν have expansions whose digits add to p − 1 in every spot.
Note that when p = 2, this characterization of Sp agrees with a well-known
description of the Sierpiński gasket fractal with vertices L, M, and N.
Theorem 6.7. If b = m, then the intersection of the boundary B and the
hyperplane X = {t ∈ R3 : ktk = 2} is the Sierpiński p-gasket Sp .
Proof. It is easy to see that the vertices L, M, and N of Sp lie in B: for instance,
L ∈ U and {(1 − p−s )L}s≥1 is a sequence of points in L converging to L. This
and Remark 5.5 allow us to restrict our attention to points u ∈ X ∩ (0, 1]3 . For
such u, we show that either both u ∈ B and u ∈ Sp , or neither of these hold.
4 As

observed in Remark 2.2, there are at most 8 possible expansions for the point v.

22

P∞
Let u = e=1 ue /pe be the unique expansion of u that is non-terminating
in each coordinate. Among all possible expansions of u, this one is of particular
interest, for by Remark 2.5,
huie =

ue
u1
+ ··· + e
p
p

(6.1)

for every e ≥ 1.
Set r = inf{e ≥ 1 : kue k =
6 2p − 2}. If r = ∞, then the componentwise
non-terminating expansion shows that u ∈ Sp , while (6.1) also shows that
khuie k = 2 − 2/pe for every e ≥ 1, and so u ∈ B by Theorem 6.5. Suppose
r < ∞. As kuk = 2 and kue k = 2p − 2, for each 1 ≤ e < r, kur k must be either
2p − 1 or 2p − 3.
If kur k = 2p − 3, then (6.1) shows both that khuie k = 2 − 2/pe for each e < r,
and that khuir k = 2 − 3/pr . By Remark 6.3, u ∈ Q3pr , and so in particular,
ue = (p − 1)1 for each e > r. This shows that khuie k = 2 − 3/pe , for each e > r
and Theorem 6.5 then implies that u ∈ B. On the other hand, taking one of the
components of u whose rth digit is not p − 1 and replacing its non-terminating
expansion with its terminating expansion, the norm of the rth digit increases
by 1, the norm of all subsequent digits decrease by p − 1, and the prior digits
remain unchanged (see Remark 2.2)—so we obtain an expansion of u whose
digits add to 2p − 2 in every spot, whence u ∈ Sp .
If kur k = 2p − 1, then (6.1) implies that khuir k = 2 − 1/pr , so u ∈
/ B, by
Theorem 6.5. We must show that no expansion of u satisfies the definition
of
the Sierpiński p-gasket. This is the case for the non-terminating expansion
P∞
e
e=1 ue /p . Any other expansion is obtained from this one by replacing the
non-terminating expansion of one or more components with the terminating
expansion (if at all possible). But Remark 2.2 shows that any such replacement,
when performed to an expansion where the first digit with norm 6= 2p − 2 has a
larger norm, will result in an expansion
P∞ with ethe same property. Consequently,
performing any such changes to
e=1 ue /p will not lead to an expansion
satisfying the definition of the Sierpiński p-gasket.
The next result, which explains the presence of Sierpiński p-gaskets in Figure 5,
relates regions attached to b to regions attached to m. We temporarily decorate
our notations for the regions attached to m with the subscript m.
Proposition 6.8. If v ∈ L ∩ N3 and kvk = deg U V − 2, then B ∩ [v, v + 1] =
v + Bm , the translation of Bm by v.
Proof. By Lemma 5.4, `v m ⊆ mdeg U V −1 ⊆ b, and yet `v ∈
/ b, since v ∈ L . This
shows that (b : `v ) = m. For each a ∈ N3 and each q we have (b[q] : `qv+a ) =
((b[q] : `qv ) : `a ) = ((b : `v )[q] : `a ) = (m[q] : `a ), where the second equality
follows from the flatness of the Frobenius over k[x, y]. Thus, `qv+a ∈ b[q] if and
only if `a ∈ m[q] or, equivalently, v + a/q ∈ U if and only if a/q ∈ Um . It
follows that U ∩ [v, ∞) ∩ Q3p∞ = (v + Um ) ∩ Q3p∞ . Taking closures we see that
U ∩ [v, ∞) = v + Um , and consequently L ∩ [v, ∞) = v + Lm . From this, it
is easy to see that v + Bm ⊆ B ∩ [v, v + 1] and B ∩ (v, v + 1] ⊆ v + Bm . To
23

complete the proof, let u ∈ B ∩ [v, v + 1] be a point with ui = vi , for some i.
As u ∈ U , the point u − v lies in Um . Since u − v ∈ [0, 1]3 and one of its
coordinates is 0, it must be the case that some other coordinate equals 1 (see
Example 3.9), and u − v ∈ Bm .
With a little more work, we can characterize the intersection of the boundary
with the trivial region, for arbitrary b = hU, V i.
Corollary 6.9. Let P = L ∩ {v ∈ N3 : kvk = deg U V − 2}. Then the
intersection the boundary B and the hyperplane X = {t ∈ R3 : ktk = deg U V }
is the Minkowski sum P + Sp = {p + s : p ∈ P and s ∈ Sp }.
Proof. Theorem 6.7 and Proposition 6.8 show that P + Sp ⊆ B ∩ X. To show
the reverse inclusion, we show that for each u ∈ B ∩ X there exists v ∈ P such
that v ≤ u ≤ v + 1. Proposition 6.8 then shows that u − v ∈ Bm , so u − v ∈ Sp ,
by Theorem 6.7, and u = v + (u − v) ∈ P + Sp .
Let u ∈ B ∩ X. Since kuk = deg U V and u ∈ B ⊆ [0, deg U V − 1]3 , u can
have at most one zero coordinate. Suppose u has one zero coordinate. If one
of the two nonzero coordinates of u is not an integer, then neither is the other
one, as kuk ∈ N; thus, buc has norm deg U V − 1, and is therefore in U , by
Lemma 5.4. This implies that u lies in the interior of U , a contradiction. So
u ∈ N3 , and subtracting 1 from each of its nonzero coordinates we obtain the
desired point in P. So we assume from now on that u has positive coordinates.
Suppose u ∈
/ N3 . As hui0 < u ∈ B, hui0 ∈ L . Furthermore, Remark 6.3
and the assumption u ∈
/ N3 show that khui0 k is either deg U V − 1 or deg U V − 2.
The first possibility, however, is ruled out by Lemma 5.4 and the fact that
hui0 ∈ L . Thus, hui0 ∈ P is the desired point.
Suppose now that u ∈ N3 . We claim that one of the points u − e1 − e2 and
u − e1 − e3 lie in L , and consequently in P. To see that, we analyze some
values of the function ∆ attached to ` and b, noting that, by Proposition 3.5(3),
t ∈ U ⇐⇒ ∆(t) = |ktk − deg U V |.
As kuk = deg U V , ∆(u) = 0. By SG 4, ∆(u − e1 ) = 1, the possible values for
∆(u − e1 − e2 ) and ∆(u − e1 − e3 ) are 0 or 2, and the possible values for ∆(u − 1)
are 1 or 3. But since u ∈ B, u − 1 ∈ L , and consequently ∆(u − 1) = 1. Thus,
SG 5 shows that one of ∆(u − e1 − e2 ) and ∆(u − e1 − e3 ) must be 0, and thus
one of u − e1 − e2 and u − e1 − e3 lie in L , establishing our claim.
Next, we turn our attention to the F -threshold function. The following result
is an immediate corollary of Theorem 5.9 and Theorem 6.5.
Corollary 6.10. Let t ∈ R3>0 , and set u = (deg U V /ktk) t. Suppose `due−1 ∈
/ b,
and set µ = inf{s ≥ 1 : khuis k = deg U V − 1/ps }.
(1) If µ = ∞, then ftb (`t ) = deg U V /ktk.
n hu i hu i hu i o
1
2
3
(2) Otherwise, ftb (`t ) = max t1 µ , t2 µ , t3 µ .
24

For the remainder of this section, for simplicity we specialize to the case
that b = m. We will be concerned with studying the value of the F -threshold
function ftm (`t ) for some fixed t ∈ R3>0 as the characteristic varies, and we will
be especially concerned with understanding when ftm (`t ) is determined by the
trivial region, that is, when ftm (`t ) = 2/ktk. Let u be the “normalized” point
u = 2t/ktk. In a fixed characteristic p, Theorem 5.9 and Theorem 6.7 show that
ftm (`t ) is determined by the trivial region if and only if u ∈ Sp .
If u ∈
/ [0, 1]3 , then u cannot lie in Sp for any p, and so ftm (`t ) can never equal
2/ktk. On the other hand, if u ∈ [0, 1]3 and some coordinate of u equals zero or
one, then u must lie on some edge of Sp , and so ftm (`t ) is always determined
by the trivial region. Thus, the interesting case to consider is when u ∈ (0, 1)3 .
However, the fractals Sp have area 0, and so such a point u rarely lies in Sp .
Thus, in some fixed characteristic, ftm (`t ) is “almost never” determined by the
trivial region. Moreover, because the union of these countably many fractals still
has area 0, for “most” t ∈ R3>0 , ftm (`t ) is not determined by the trivial region
in any characteristic. The following result then comes as a surprise:
Corollary 6.11. Fix a point a = (A, B, C) ∈ N3 with positive coordinates, set
D = kak, and suppose that the normalized point u = 2a/D lies in (0, 1)3 . Fix a
prime p that does not divide D, and let O denote the multiplicative order of p
modulo D. If s ≥ 1, then


[2Aps % D] [2Bps % D] [2Cps % D]
zs =
,
,
D
D
D
lies in (0, 1)3 , and has coordinate sum equal to either 1 or 2.
(1) If kzs k = 2 for all 1 ≤ s ≤ O, then ftm (`a ) = 2/D.
(2) Otherwise, if 1 ≤ µ ≤ O is the minimal integer for which kzµ k = 1, then


[2Apµ % D] [2Bpµ % D] [2Cpµ % D]
2
1
m a
ft (` ) =
−
· min
,
,
.
D pµ
AD
BD
CD
In particular, ftm (`a ) = 2/D for all p ≡ 1 mod D.
Proof. By Remark 2.6,

zs
,
(6.2)
ps
and Remark 6.3 and the assumption that p - D then imply that zs ∈ (0, 1)3 ,
and that either kzs k = 1 or kzs k = 2. If kzs k = 2 for every 1 ≤ s ≤ O, then
the same holds for all s ≥ 1. In this case, (6.2) and Corollary 6.10 imply that
ftm (`t ) = 2/D. Otherwise, Corollary 6.10 instead implies that
( 2A
)
2B
2C
huis = u −

ftm (`a ) = max

D

µ

A

,

D

B

µ

,

D

C

µ

.

The claimed formula for ftm (`a ) follows from applying Remark 2.6 to simplify
this expression, and the last assertion is justified by the observation that if
p ≡ 1 mod D, then zs = u for all s ≥ 1.
25

Remark 6.12 (F -pure thresholds as a function of the class of p). In the statement
of Corollary 6.11, the value of µ depends only on the class of p modulo kak.
Thus, is this context, Corollary 6.11 shows that for every unit u modulo kak,
there exist a positive integer µ(u) and a nonnegative integer E(u) such that
ftm (`a ) =

2
E(u)
− µ(u)
kak p

whenever p ≡ u mod kak.
Though we will not elaborate on this here, this description of the F -pure
threshold of the polynomial `a , when combined with certain well-known facts
(see, e.g., [MTW05, Proposition 1.9] or [Her12, Key Lemma 3.1]) allows us to
give a positive answer to [MTW05, Problem 3.10] in this setting.

7

F -pure thresholds of homogeneous polynomials in two variables

Let G be a non-constant form in k[x, y] and b = hU, V i, where U, V ∈ k[x, y] are
non-constant relatively prime forms. Extending k, if necessary, we write G = `a ,
where ` = (`1 , . . . , `n ) is a collection of pairwise prime linear forms and a ∈ Nn>0 .
Let λ = deg U V / deg G. As before, we shall say that “ftb (G) is determined by a
critical point c” if ftb (G) = max{c1 /a1 , . . . , cn /an } < λ or, equivalently, c < λa;
see Proposition 5.3 and Theorem 5.9. According to Theorem 5.9, we have three
mutually exclusive possibilities for ftb (G) = ftb (`a ):
(A) ftb (G) is determined by a critical point in Nn . This is the case if and only
if hλai0 ∈ U .
(B) ftb (G) is determined by a critical point not in Nn . This is the case if and
only if hλai0 ∈
/ U but hλaie ∈ U for some e ≥ 1.
(C) ftb (G) is determined by the trivial region: ftb (G) = λ.
Remark 7.1. In case (B), taking q = pe > 1 to be the least power of p such that
qc ∈ Nn , Remark 5.11 shows that
0 < λ − ftb (G) ≤

n−2
1
< .
q deg G
q

If ftb (G) ∈ Qq (which will be the case, e.g., when G is square free), then the
above inequalities and Remark 2.4 show that ftb (G) = hλie . This reproduces a
result obtained by Núñez-Betancourt, Witt, Zhang, and the first author, through
completely different methods [HNWZ16]. For more on this, see Theorem 8.12.
Example 7.2. In the above remark, the conclusion that ftb (G) is a truncation
of λ does not hold under the looser assumption that ftb (G) ∈ Qp∞ . If G =
(xy)49 ((x + y)(x + 2y)(x + 4y))13 ∈ F7 [x, y] and m = hx, yi, for instance, then
ftm (G) = 4/343 ∈ Q73 (determined by the critical point (4, 4, 1, 1, 1)/7 associated
with ` = (x, y, x + y, x + 2y, x + 4y) and m), while hλi3 = h2/137i3 = 5/343.
26

We now specialize to the case where b = m = hx, yi. Recall that ftm (G) is
the F -pure threshold of G, denoted by fpt(G).
Remark 7.3. Proposition 5.6(4) shows that the only critical points with integer
coordinates associated with ` and m are e1 , . . . , en . Moreover, those are the
only critical points that have some positive integer coordinate. Indeed, if c is a
critical point and ci ∈ N>0 , then c ≥ ei , and therefore c = ei .

The scarcity of positive integer coordinates in critical points observed above
has interesting consequences. Case (A) becomes relegated to a “degenerate case”
where the multiplicity of some `i in G is too large—fpt(G) is determined by
ei if and only if λa = (2/ deg G)a > ei if and only if ai > (deg G)/2, in which
case fpt(G) = max{0, 1/ai } = 1/ai . Note that the same conclusion is reached if
ai = (deg G)/2 (so λa ≥ ei ), but in that case fpt(G) = 1/ai = λ, so this falls
actually under case (C).
In case (B), no nonzero coordinate of the critical point c is integral, so the
minimal denominator of fpt(G) = max{ci /ai } is of the form kpe , with e ≥ 1 and
k a factor of one of the multiplicities ai . To put this observation in context, we
must digress momentarily with some characteristic 0 considerations.
Definition 7.4. Let G0 ∈ Q[x, y] be a non-constant form. We say that a prime p
is a good prime associated with G0 if there exists a reduction modulo p of G0 in
Fp [x, y], which we denote by Gp , and the factorization of Gp over Fp is similar
to the factorization of G0 over C, in the sense that those factorizations have the
same number of pairwise prime linear factors and the same multiplicities. If that
is not the case, we say that p is a bad prime.
Lemma 7.5. There are at most finitely many bad primes associated with a fixed
non-constant form G0 ∈ Q[x, y].
Proof. The proof relies on the following facts. Given a finitely generated Zalgebra A containing Z, for all but finitely many primes p there exists a maximal
ideal of A containing p. Moreover, if M is a maximal ideal of A containing a
prime number p, then A/M is a finite field of characteristic p. For a justification
of these facts, see, e.g., [Her11, Corollary 3.2].
We now proceed with the proof. Suppose G0 factors over C as
G0 = γ · xl y m (x − α1 y)k1 · · · (x − αr y)kr ,

where γ, α1 , . . . , αr ∈ C× and the αi are Q
distinct. Let A be the Z-algebra
generated by γ and the αi , together with i<j (αi − αj )−1 · (γ · α1 · · · αr )−1 .
According to the facts cited above, given a prime p  0, there exists a maximal
ideal M of A containing p. Furthermore, for such p and M, the field k = A/M
is a finite extension of Fp over which the image of G0 has a factorization similar
to that of G0 , because, by design, the images of γ and the αi and αi − αj in k
are nonzero, since they are invertible. So p is a good prime.
Definition 7.6. A form F in two variables over some field K is degenerate, of
degeneracy type m, if it has a linear factor over K with multiplicity m > (deg F )/2.
Note that each degenerate form has a unique degeneracy type.
27

Let G0 ∈ Q[x, y] be a non-constant form. As discussed in the introduction, the
log canonical threshold of G0 , denoted by lct(G0 ), is an invariant measuring the
singularity of G0 at the origin, and is defined via a log resolution of singularities.
In the context of this paper, the most important property of lct(G0 ) is that
limp→∞ fpt(Gp ) = lct(G0 ).
If G0 is degenerate, of degeneracy type m, then for each good prime p the
computation of fpt(Gp ) falls under case (A), and fpt(Gp ) = 1/m. Consequently,
lct(G0 ) = limp→∞ fpt(Gp ) = 1/m. If G0 is non-degenerate, then for each
good prime p the computation of fpt(Gp ) falls under cases (B) or (C), and
the inequalities in Remark 7.1 show that lct(G0 ) = limp→∞ fpt(Gp ) = λ =
2/ deg(G0 ). Thus, in the paragraph before Definition 7.4 we have shown the
following result, which provides an affirmative answer to Question 1.1 in the
two-variable homogeneous setting.
Theorem 7.7. Let G0 ∈ Q[x, y] be a non-constant form. Let p be a good prime
associated with G0 , and let Gp be the image of G0 in Fp [x, y]. If fpt(Gp ) 6=
lct(G0 ), then the minimal denominator of fpt(Gp ) is of the form kpe , where
e ≥ 1 and k divides the multiplicity of some linear factor (over C) of G0 .
Remark 7.8. If we weaken the notion of good prime, requiring only that G0
and Gp be both non-degenerate or both degenerate, of the same degeneracy type,
then an alternate version of the above theorem still holds, where the conclusion
states that k divides the multiplicity of some linear factor (over Fp ) of Gp .
Example 7.9. Let G0 = x(x + y)(x + 6y). Then G5 = x(x + y)2 , so fpt(G5 ) =
1/2 6= 2/3 = lct(G0 ), and yet fpt(G5 ) has a denominator prime to 5. This shows
the need for requiring p to be a good prime in Theorem 7.7. That result may
otherwise not hold when the factorizations of G0 and Gp are “too different”.
Going back to the three cases discussed earlier in this section, while case (A)
is clearly delimited, distinguishing between cases (B) and (C) is delicate. For
that, it is useful to know an upper bound for the denominator of the critical
point that determines fpt(G) in case (B). When deg G is prime to p we have
such a bound—a consequence of the “forbidden intervals” theorem of Blickle,
Mustaţă, and Smith [BMS09, Proposition 4.3], generalized by the first author
[Her12, Proposition 4.8], which states that for any β ∈ (0, 1)q there are no F -pure
thresholds of hypersurfaces in characteristic p in the interval (β, βq/(q − 1)).
Lemma 7.10. Let λ = a/b ∈ (0, 1] ∩ Q. Suppose b is prime to p, and let µ be the
multiplicative order of p in (Z/bZ)× . Then no F -pure threshold of a polynomial
over a field of characteristic p lies in the interval (hλiµ , λ).
Proof. Let q = pµ and k = (q − 1)λ; then k ∈ N and


k
k
1
1
λ=
=
1 + + 2 + ··· .
q−1
q
q
q
Since k < q, the above equation shows that hλiµ = k/q, so λ = hλiµ q/(q − 1),
and the “forbidden intervals” theorem gives the result.
28

Proposition 7.11. Let G ∈ k[x, y] be a form of degree d > 0. Suppose
fpt(G) < λ = 2/d and the minimal denominator of λ is prime to p. Let µ
be the multiplicative order of p modulo that denominator. Then fpt(G) is determined by a critical point with coordinates in Qpµ .
Proof. Write G = `a , as in the beginning of this section, and let c be the critical
point that determines fpt(G). Lemma 7.10 gives us the following inequalities:


c1
cn
2
max
,...,
= fpt(G) ≤ hλiµ < λ = .
a1
an
d
2
As max{ci /ai } · a ∈ ∂[c, ∞) and d2 · a = kak
· a ∈ ∂T , multiplying each of
the terms in the above inequalities by a shows that hλiµ a lies in B∆(c) (c), the
region where the function ∆ attached to ` and m is determined by c. Because
hλiµ a ∈ Qnpµ , Remark 2.16 tells us that c ∈ Qnpµ as well.

We summarize the above observations in the following theorem.
Theorem 7.12. Let G ∈ k[x, y] be a form of degree d > 0. Write G = `a1 1 · · · `ann ,
where the `i are pairwise prime linear forms in k[x, y] and ai > 0, for each i.
(1) If ai ≥ d/2, for some i, then fpt(G) = 1/ai .
(2) Suppose ai < d/2, for each i. Set ` = (`1 , . . . , `n ) and a = (a1 , . . . , an ).
Then exactly one of the following holds:
• d2 · a e lies in the upper region attached to ` and m = hx, yi, for some
e ≥ 1. Thus, there exists a unique critical point c ∈ Qnpe associated with `
and m such that c ≤ d2 · a e , and


c1
cn
,...,
fpt(G) = max
.
a1
an
The critical point c has no nonzero integer coordinates, and thus the minimal
denominator of fpt(G) is of the form kpm , where 1 ≤ m ≤ e and k is a
factor of some multiplicity ai . Moreover, if the minimal denominator of
2/d is prime to p, then the above holds for some e no greater than the
multiplicative order of p modulo that denominator. Finally, if fpt(G) ∈ Qpe
(e.g., if G is square free), then fpt(G) = h2/die .
• fpt(G) = 2/d.

Example 7.13. Consider the forms G1 = x2 y 2 (x2 + 2xy + 3y 2 )7 and G2 =
x2 y 2 (x2 + 2xy + 3y 2 ) in F25 [x, y]. Let `1 = x, `2 = y, and `3 `4 = x2 + 2xy +
3y 2 , and set ` = (`1 , `2 , `3 , `4 ). Finally, let a1 = (2, 2, 7, 7), a2 = (2, 2, 1, 1),
λ1 = 2/ deg G1 = 1/9, and λ2 = 2/ deg G2 = 1/3. Since the multiplicative
order of 5 (mod 9) is 6, to find fpt(G1 ) we look for a truncation hλ1 a1 ie with
e ≤ 6 that lies in the upper region U attached to ` and m. We find that
hλ1 a1 i3 = (27, 27, 97, 97)/125 lies in U , and is itself a critical point. Thus
29

 27
97
97
fpt(G1 ) = max 125·2
= 875
. As for G2 , the multiplicative order of 5
, 125·7
(mod 3) is 2, but hλ2 a2 i2 does not lie in U , so fpt(G2 ) is determined by the
trivial region: fpt(G2 ) = λ2 = 1/3. Figure 6 shows a density plot of a section
of the function ∆ attached to ` and m, together with the lines spanned by a1
and a2 .
The method used in the above example can be applied to any form in two
variables, and leads to an efficient algorithm that has been implemented by the
second author in the Macaulay2 [GS] package PosChar [BBH+ ]. More details
about this algorithm and its implementation are presented in Appendix A.
We close this section highlighting a key difference between F -pure thresholds
and F -thresholds with respect to ideals b 6= m, which is that in the latter setting
the analogue of Remark 7.3 does not hold. Integral critical points tend to
abound (see, for instance, Example 4.7) and, as the following example will show,
critical points may have both nonzero integral coordinates and non-integral
coordinates—so F -thresholds with a denominator prime to p may arise from
non-integral critical points.
Example 7.14. Let ` = (x, y, x+y, x+2y) ∈ F5 [x, y]4 and b = hx, y 2 i ⊆ F5 [x, y].
Then c = (2/5, 3/5, 4/5, 1) is a critical point associated with ` and b, and it is
easy to produce forms whose F -thresholds with respect to b are determined by
the last coordinate of c and have a denominatorprime to 5. For instance, if
2
3
4
1
1
G = x7 y 10 (x + y)13 (x + 2y)16 , then ftb (G) = max 5·7
, 5·10
, 5·13
, 16
= 16
.

8

F -pure thresholds of quasi-homogeneous polynomials in two variables

If K is an arbitrary field, we endow K[X, Y ] with a non-standard N-grading where
deg X = u, deg Y = v, and uv 6= 0. We shall refer to the homogeneous elements
under this grading as quasi-homogeneous polynomials and reserve the terms
form and homogeneous polynomial for polynomials that are homogeneous under
the standard grading. We extend the notion of degeneracy (see Definition 7.6)
to this setting by saying that a quasi-homogeneous polynomial f ∈ K[X, Y ] is
degenerate, of degeneracy type m, if it has an irreducible factor over K with
multiplicity m > deg(f )/(u + v).
Notation. Given f ∈ K[X, Y ] and an irreducible polynomial h ∈ K[X, Y ],
multh (f ) denotes the multiplicity of h in f , that is, the largest m ∈ N (possibly 0)
such that hm divides f in K[X, Y ].
Definition 8.1. Let g0 ∈ Q[X, Y ] be a non-constant quasi-homogeneous polynomial. A prime p is a good prime associated with g0 if the following hold:
(1) there exists a reduction modulo p of g0 in Fp [X, Y ], denoted by gp ;
(2) the factorization of gp over Fp is similar to the factorization of g0 over C, in
the sense that those factorizations have the same number of pairwise prime
irreducible factors and the same multiplicities;
30

1

0.5

0
0

0.5

1

Figure 6: The two dimensional section (u, v) 7→ ∆(u, u, v, v) of ∆ and the lines
spanned by a1 and a2

31

(3) multX (gp ) = multX (g0 ) and multY (gp ) = multY (g0 ).
If that is not the case, then p is a bad prime. Remark 8.6 will show that there
exist at most finitely many bad primes associated with a fixed g0 .
Our first and main goal in this section will be to extend Theorem 7.7 to the
quasi-homogeneous setting:
Theorem 8.2. Let g0 ∈ Q[X, Y ] be a non-constant quasi-homogeneous polynomial. Let p be a good prime associated with g0 , and let gp be the image of g0 in
Fp [X, Y ]. If fpt(gp ) 6= lct(g0 ), then the minimal denominator of fpt(gp ) is of the
form kpe , where e ≥ 1 and k is a factor of one of the following: multX (g0 )·deg X,
multY (g0 ) · deg Y , or multh (g0 ), where h is some irreducible factor (over C) of
g0 other than X or Y .
Remark 8.3. As it is the case with Theorem 7.7 (see Remark 7.8), an alternate
version of the above result, where, in the conclusion, multiplicities in g0 are
replaced with multiplicities in gp , may be obtained under a looser notion of
good prime, that only requires that g0 and gp be both non-degenerate or both
degenerate, of the same degeneracy type.
Fix g0 as in the statement of Theorem 8.2 and a good prime p; let k = Fp
and g = gp , the image of g0 in k[X, Y ]. Since Theorem 8.2 is already known for
standard homogenous polynomials, we may assume that g0 is not homogeneous—
so g0 is not a monomial, and thus neither is g, and u 6= v. In fact we assume,
by possibly changing the grading, that u and v are coprime. The following
proposition will allow us to extend our methods to the quasi-homogeneous
setting:
Proposition 8.4. Let ψ : k[X, Y ] → k[x, y] be the map f (X, Y ) 7→ f (xu , y v ).
Set G = ψ(g) and b = hxu , y v i. Then fpt(g) = ftb (G).
Proof. Because of the description of the F -threshold of a polynomial given in
Discussion 4.1, it suffices to show that, for each a ∈ N and q = pe , we have that
[q]
g a ∈ hX, Y i if and only if Ga ∈ b[q] . Let A = k[xu , y v ] and B = k[x, y]. Let a
be the ideal of A generated by xu and y v . Then ψ induces an isomorphism from
k[X, Y ] to A, mapping g to G and m = hX, Y i to a, and hence, g a ∈ m[q] if
and only if Ga ∈ a[q] . But A is a direct summand of B as an A-module, and
therefore b[q] ∩ A = (a[q] B) ∩ A = a[q] . As G ∈ A, we see that Ga ∈ a[q] if and
only if Ga ∈ b[q] , which allows us to conclude the proof.
Lemma 8.5. The polynomial g can be factored as
g = ξ · X j1 Y j2 ·

m
Y

(X v − µi Y u )ki ,

i=1

for some j1 , j2 ∈ N, m, k1 , . . . , km ∈ N>0 , ξ ∈ k× , and distinct µ1 , . . . , µm ∈ k× .
Remark 8.6. The polynomial g0 , of course, has a similar factorization over C,
and the argument used in the proof of Lemma 7.5 can be adapted to show that
there exist at most finitely many bad primes associated with g0 .
32

Proof. Write g as X j1 Y j2 h, where h is a quasi-homogeneous polynomial prime
to XY . As g is not a monomial, h has at least terms X a and Y b , with au = bv.
Since u and v are coprime, v | a and u | b. Now suppose h also has a term X c Y d .
Then cu + dv = au, so v | c − a; but v | a as well, so we conclude that v | c.
Similarly we find that u | d, so in each term of h the exponents of X and Y
are divisible by v and u, respectively, and therefore h = H(X v , Y u ) for some
form H. The result is then obtained by factoring H into linear forms.
In the remainder of this section, we adopt the notation introduced in Proposition 8.4. In our proof of Theorem 8.2, we assume that j1 j2 6= 0; our argument
can be adapted to handle the other cases, which are simpler. As u = deg X and
v = deg Y are coprime, p can only divide one of them; we assume that u is prime
to p and write v = wq̄, where w is prime to p and q̄ is a power of p (possibly 1).
1/q̄
1/q̄ q̄
Then, as k = k, there exist unique µi ∈ k with µi
= µi , so that
G = ψ(g) = ξ · xuj1 y vj2 ·

m 
Y
i=1

1/q̄

xuw − µi y uw

q̄ki

,

(8.1)

1/q̄

where the factors xuw − µi y uw are square free and pairwise prime. As our
methods will depend on factoring G into a product of linear forms, it will be
1/q̄
necessary to factor each xuw − µi y uw into a product of linear forms. Let ζ be
1/q̄
a primitive (uw)th root of unity in k, and let νi be a (uw)th root of µi in k,
Q
1/q̄
uw
for each i; then xuw − µi y uw = j=1 (x − νi ζ j y). Substituting this into (8.1)
produces the following factorization of G into a product of linear forms:
G=ξ·x

uj1 vj2

y

·

m Y
uw
Y
i=1 j=1

(x − νi ζ j y)q̄ki .

Let n = 2 + muw, the number of pairwise prime linear factors of G. It will be
convenient to label the n linear factors of G and the canonical basis vectors of
Rn in a non-standard way. Let `1 = x, `2 = y, and for each i ∈ {1, . . . , m} and
j ∈ {1, . . . , uw} let `i,j = x − νi ζ j y; set
` = (`1 , `2 , `1,1 , . . . , `1,uw , . . . , `m,1 , . . . , `m,uw ).
For each `i,j , let fi,j be the corresponding canonical basis vector of Rn , and set
Puw
Quw
1/q̄
fi = j=1 fi,j , so that `fi = j=1 (x − νi ζ j y) = xuw − µi y uw , by definition.
The canonical basis vectors corresponding to `1 and `2 will be denoted by the
usual e1 and e2 .
We now explore some symmetries in the critical points associated with ` and
b = hxu , y v i coming from the special shape of G.
P
Lemma 8.7. Suppose c = c1 e1 + c2 e2 + i,j ci,j fi,j is a critical point. Let c0 be
the point obtained from c by replacing each set of coordinates ci,1 , ci,2 , . . . , ci,uw
with ci,uw , ci,1 , . . . , ci,uw−1 . Then c0 is also a critical point.

33

Proof. Suppose c ∈ Qnq . Then the k-automorphism of k[x, y] that maps x 7→ x
0
and y 7→ ζy transforms `qc into a constant multiple of `qc , while fixing b.
Pm
Lemma 8.8. Let t = t1 e1 + t2 e2 + i=1 t∗i fP
i ∈ ∂T . Suppose c is a critical
m
point such that c < t. Then c = c1 e1 + c2 e2 + i=1 c∗i fi , for some ci , c∗i ∈ Qp∞ .
P
Proof. Write c = c1 e1 + c2 e2 + i,j ci,j fi,j , and construct c0 as in Lemma 8.7.
Then c0 is a critical point and c0 < t as well. Since there can be no more than
one critical point lying under a point in ∂T (see Remark 5.8), we conclude that
c = c0 . Iterating this process, we see that for each i ∈ {1, . . . , m} the coordinates
ci,1 , . . . , ci,uw of c are all equal, and the result follows.
Pm
Lemma 8.9. Suppose c = αe1 + βe2 + i=1 γi fi ∈ Qnq is a critical point. Then
α/u and β/w both lie in Qq .
Proof. Suppose α 6= 0. Write α = a/q, β = b/q, and γi = ci /q. As c ∈ U ,
`qc = xa y b ·

m 
Y
i=1

1/q̄

xuw − µi y uw

ci

∈ b[q] = hxuq , y vq i.

(8.2)

Since c is a critical point, there is a monomial M = xa+iuw y b+juw in the support
of `qc such that M/x ∈
/ b[q] . Clearly a + iuw ≤ uq, and if the inequality were
strict, then (8.2) would imply that b + juw ≥ vq, and M/x would be in b[q] , a
contradiction. So a + iuw = uq, whence α/u = a/(uq) = 1 − iw/q ∈ Qq . The
argument showing that β/w ∈ Qq is analogous.
Pm
Corollary 8.10. Suppose c = αe1 + βe2 + i=1 γi fi is a critical point.
(1) If α is a positive integer, then c = ue1 .
(2) If β is a positive integer, then either c = ve2 or q̄ - β.
Proof. If α ∈ N>0 , then, bearing in mind that u is prime to p, Lemma 8.9 shows
that u | α. Thus, c ≥ ue1 , and since ue1 is a critical point we must have c = ue1 .
Likewise, if β ∈ N>0 , then w | β. If q̄ | β as well, then v = wq̄ | β, and arguing
as before we conclude that c = ve2 .
We are now ready to conclude the proof of Theorem 8.2.
Proof of Theorem 8.2. Recall that fpt(g) = ftb (G), by PropositionP8.4. Set
m
λ = (u + v)/ deg G and write G = ξ`a , where a = uj1 e1 + vj2 e2 + i=1 q̄ki fi .
Consider the following cases:
• λki > 1 for some i. This situation is not common—λki > 1 is equivalent to
deg G < (u + v)ki and, as deg G ≥ uvki , this requires either u or
 v to be 1. In
this case, q̄fi is a critical point lying under λa, so fpt(g) = max 0, q̄kq̄ i = k1i .
• λji > 1 for some i. Suppose λj1 > 1 (the othercase is analogous). Then
λa lies above the critical point ue1 , so fpt(g) = max uju1 , 0 = j11 .
34

• λji ≤ 1 and λki ≤ 1, for each i. P
If ftb (G) is determined by a critical
m
point c, then c < λa, so c = αe1 + βe2 + i=1 γi fi , for some α, β, γi ∈ Qp∞ , by
Lemma 8.8. The inequalities λji ≤ 1 (i = 1, 2) ensure that c is neither ue1 nor
ve2 , and thus Corollary 8.10 shows that α is not a positive integer, and that if β
is a positive integer, then q̄ - β. The inequalities λki ≤ 1 (i = 1, . . . , m), on the
other hand, ensure that γi < λq̄ki ≤ q̄. Being determined by c, fpt(g) = ftb (G)
equals the maximum among α/(uj1 ), β/(q̄wj2 ), and γi /(q̄ki ) (i = 1, . . . , m),
hence its minimal denominator has the desired form. Alternatively, ftb (G) may
be determined by the trivial region, and fpt(g) = ftb (G) = λ.
We now allow p to vary, to find lct(g0 ) in each of the above cases. Note that
the conditions “λji > 1” and “λki > 1” are equivalent to degeneracy conditions
on gp , which are inherited from g0 , and thus independent of the choice of the good
prime p. It follows that lct(g0 ) = fpt(gp ) in the first two cases. As for the last
case, note that if p - v then α and β cannot be nonzero integers, by Corollary 8.10,
and γi < q̄ = 1, so c ∈
/ Nn . Thus, fpt(gp ) is either λ or is determined by a
non-integral critical point, for all p  0, and the inequalities in Remark 7.1
show that lct(g0 ) = limp→∞ fpt(gp ) = λ. So we have shown that the minimal
denominator of fpt(gp ) has the required form whenever fpt(gp ) 6= lct(g0 ).
A byproduct of our proof is the following:
Corollary 8.11. If g0 is non-degenerate, then lct(g0 ) = (u + v)/ deg(g0 ). If g0
is degenerate, of degeneracy type m, then lct(g0 ) = 1/m.
The following theorem was recently proved by Núñez-Betancourt, Witt,
Zhang, and the first author.
Theorem 8.12 ([HNWZ16, Theorem 4.4]). Let g ∈ k[X, Y ] be a non-constant
quasi-homogeneous polynomial that is square free over k . Set λ = (u + v)/ deg g.
Then either fpt(g) = min{1, λ} or fpt(g) = hλie , for some e ≥ 1.
We conclude this section and the paper with a simple proof of this theorem
under the additional assumption that u and v are prime to p.
Proof. In view of Theorem 7.12, we may assume that g is not homogeneous—so
g is not a monomial and u 6= v. As before, we assume that u and v are coprime
and k = k. Using Lemma 8.5 and the assumption that g is square free, we write
G = ψ(g) = ξ · xju y kv ·

m
Y
i=1

(xuv − µi y uv ),

where j, k ∈ {0, 1} and the factors xuv − µi y uv are square free and pairwise
prime. We consider the case j = k = 1; the other cases are analogous. Set
λ = (u+v)/ deg G and b = hxu , y v i and, adopting the setup used earlier
(minding
Pm
that here v = w and q̄ = 1), write G = ξ`a , where a = ue1 + ve2 + i=1 fi .

35

If ftb (G) 6= λ, then ftb (G) is determined by a critical point c < λa. Since
λ < 1, P
we find that c 6∈ Nn ,5 and Lemmata 8.8 and 8.9 show that c = αe1 +
m
βe2 + i=1 γi fi for some α, β, γi ∈ Qq , where q = pe > 1, and ftb (G) =
max{α/u, β/v, γi } ∈ Qq . Remark 7.1 then shows that ftb (G) = hλie . As
fpt(g) = ftb (G), by Proposition 8.4, we have shown that either fpt(g) = λ =
min{1, λ} or fpt(g) = hλie , for some e ≥ 1.

Acknowledgements
This paper was written while the second author was visiting the University of
Utah, during a sabbatical leave. He wishes to thank the University of Utah for
the hospitality and inspiring environment, and Anurag Singh for making this
happen. The first author gratefully acknowledges support from the NSF through
a Mathematical Sciences Research Postdoctoral Fellowship. The authors would
like to thank the anonymous referees for the thorough reading and the valuable
comments, corrections, and suggestions.

A

The algorithm

In this appendix we present an algorithm to compute F -pure thresholds of
forms in two variables and discuss some of the practical issues surrounding its
implementation. The first issue one faces is factoring the form: this factorization
often happens in very large field extensions that cannot be handled by the
computer. We dodge this issue here, assuming that a factorization is known
from the start: G = `a1 1 . . . `ann = `a . A naı̈ve use of Theorem 7.12 then leads to
Algorithm 1.

Algorithm 1: Naı̈ve FPT Algorithm
Input: ` = (`1 , . . . , `n ), a = (a1 , . . . , an )
Output: The F -pure threshold of G = `a1 1 . . . `ann
1
2
3
4
5
6
7

if ai ≥ kak/2, for some i then return 1/ai
λ ← 2/kak
if fpt(G) = λ then return λ
e←1
while hλaie 6∈ U do e ← e + 1
Locate the critical point c ∈ Qnpe such that c ≤ hλaie

return max ac11 , . . . , acnn

5 Note that if x or y are not factors of g (i.e., jk = 0), then λ may—in some rare instances—
be ≥ 1, in which case ftb (G) is determined by an integral critical point and ftb (G) = 1.

36

Remark A.1. A few remarks are in order:
• In line 3 we test if fpt(G) = λ. Testing whether fpt(G) equals any given rational
number can be done rather efficiently by a method of Schwede, which compares
the “non-F -pure ideals” of [FST11] and test ideals. This was implemented
by Schwede in the Macaulay2 package PosChar [BBH+ ], in the command
isFPTPoly.
• Checking if hλaie ∈ U in line 5 boils down to checking ideal membership
e
e
(specifically, `p hλaie ∈ m[p ] ), so it can be easily implemented.
• The ‘while’ loop in line 5 is guaranteed to end, by Theorem 7.12(2).
• In line 6, we search for a critical point c ≤ hλaie , guaranteed to exist by Theorem 7.12(2). This is done in the most naı̈ve way, by successively subtracting
1/pe from the coordinates of hλaie , until a minimal point of U ∩ Qnpe is found.
See Proposition 5.6 and the comments after its proof.
Discussion A.2. Although there is no question that Algorithm 1 works in theory,
it does not fare well in practice. Directly checking if hλaie ∈ U (i.e., checking if
e
e
`p hλaie ∈ m[p ] ) for increasingly large e (line 5) can quickly lead to impractical
computations involving polynomials of extremely large degrees. To get around
this problem, write the componentwise non-terminating base p expansion of λa:
λa =

d2
d3
d1
+ 2 + 3 + ··· .
p
p
p

[p]
Set b0 := m, and successively compute be := (be−1 : `de ). Using the flatness of
the Frobenius over k[x, y] we see that
e

be = (m[p ] : `p

e−1

d1 +pe−2 d2 +···+de

e

e

) = (m[p ] : `p

hλaie

),

so be = h1i if and only if hλaie ∈ U . When computing the ideals be we never
raise polynomials to powers greater than p. Moreover, it can be shown that each
be can be generated by two forms whose degrees add up to at most n. So we
can check whether hλaie ∈ U for arbitrarily large e without ever having to deal
with large degree polynomials.
Searching for a critical point c ≤ hλaie is also impractical if e is large. In
our improved algorithm, Algorithm 2, we use the above ideas in the search for
critical points, relying on the next lemma, which relates regions and critical
points with respect to different ideals.
To simplify our language, in what follows we shall refer to a “critical point
associated with ` and b” simply as a “b-critical point”, and denote the upper
region attached to ` and b by Ub , unless b = m, in which case we shall simply
denote the upper region by the usual U . We shall also use the term “critical
point under u” for a critical point c such that c ≤ u.
Lemma A.3. Let b be an ideal generated by two non-constant relatively prime
forms in k[x, y]. Let q be a power of p, k ∈ Nn , and b0 = (b[q] : `k ). Finally, let
u ∈ (Q≥0 )np∞ and v = (u + k)/q. Then:
37

(1) u ∈ Ub0

⇐⇒ v ∈ Ub .

(2) If v is a b-critical point, then u is a b0 -critical point. The converse holds if
ui > 0 whenever vi > 0, and in particular when u has positive coordinates.
0

0

0

Proof. The first point follows from the fact that (b[q] : `k )[q ] = (b[qq ] : `q k ), due
to the flatness of the Frobenius over k[x, y]. The second point follows from the
first and the characterization of critical points given in Proposition 5.6(3).
The following corollary shows how the above lemma will be used in Algorithm 2.
Corollary A.4. Adopt the notation introduced in Discussion A.2. Suppose
be = h1i, so that hλaie ∈ U . For each j ∈ N with j < e, let
uj = pj (hλaie − hλaij ) =

dj+1
dj+2
de
+ 2 + · · · + e−j .
p
p
p

Then:
(1) uj ∈ Ubj and, in particular, ue−1 = de /p ∈ Ube−1 .
(2) If cj ∈ Qnpe−j is a bj -critical point under uj with positive coordinates, then
cj /pj + hλaij is an m-critical point under hλaie .
j

Proof. Set k = pj hλaij . Then hλaie = (uj + k)/pj and (m[p ] : `k ) = bj , and
the corollary follows easily from Lemma A.3.
We are now ready to present our improved algorithm—see Algorithm 2 on
page 39—and prove its correctness.
Theorem A.5. Algorithm 2 works.
Proof. The algorithm may terminate prematurely in lines 1, 5, or 13. If it
terminates in line 1, then it returns the correct output, by Theorem 7.12(1). If
it terminates in line 5, then it returns the correct output for obvious reasons. If
the algorithm gets past line 6, then at that point λ = 2/ deg(G) and m is the
minimal denominator of λ, and one of the following holds:
• p | m, fpt(G) 6= λ, and µ = ∞;
• p - m, and µ is the multiplicative order of p modulo m; it could still be the
case that fpt(G) = λ.
The ‘while’ loop in line 9 will then compute the ideals be introduced in Discussion A.2, until be = h1i (i.e., hλaie ∈ U ) or e = µ. This loop terminates—this is
clear if µ < ∞, and if µ = ∞, then we know that fpt(G) 6= λ, so some truncation
hλaie lies in U , by Theorem 7.12(2). If be 6= h1i at the end of this loop, the
aforementioned theorem allows us to conclude that fpt(G) = λ; the algorithm
terminates in line 13, returning the correct output.

38

Algorithm 2: Improved FPT Algorithm
Input: ` = (`1 , . . . , `n ), a = (a1 , . . . , an )
Output: The F -pure threshold of G = `a1 1 . . . `ann
1
2
3
4
5
6
7
8
9
10
11

12
13
14
15
16
17
18
19
20
21
22

if ai ≥ kak/2, for some i then return 1/ai
λ ← 2/kak
m ← minimal denominator of λ
if p | m then
if fpt(G) = λ then return λ else µ ← ∞

else µ ← multiplicative order of p modulo m
b0 ← m
e←0
while be 6= h1i and e < µ do
e←e+1
de ← eth digit of the unique componentwise non-terminating base
p expansion of λa
[p]

be ← (be−1 : `de )

if be 6= h1i then return λ
j ←e−1
vj ← de /p
cj ← bj -critical point under vj
while some component of cj is 0 and j > 0 do
vj−1 ← (cj + dj )/p
j ←j−1
cj ← bj -critical point under vj
c ← cj /pj + hλaij

return max ac11 , . . . , acnn

39

Suppose the algorithm gets past line 13. At that point, we have hλaie ∈ U , so
ve−1 = de /p ∈ Ube−1 , by Corollary A.4(1). In line 16 we look for a be−1 -critical
point ce−1 ∈ Qnp under ve−1 . If ce−1 has positive coordinates (so the ‘while’
loop in line 17 is bypassed), then c = ce−1 /pe−1 + hλaie−1 , computed in line 21,
is an m-critical point under hλaie , by Corollary A.4(2), so that in this situation
the algorithm returns the correct output.
It remains to examine what happens when ce−1 has some zero coordinate. In
the ‘while’ loop in line 17, points vj and cj (j = e − 2, e − 3, . . .) are constructed
so that vj−1 = (cj + dj )/p and cj is a bj -critical point under vj . We claim that,
for each 0 ≤ j < e, vj ∈ Ubj (so it makes sense to look for a critical point cj
under vj in line 20) and vj ≤ uj := pj (hλaie − hλaij ). This follows from an easy
inductive argument using Lemma A.3 and the recursions uj−1 = (uj + dj )/p
[p]
and bj = (bj−1 : `dj ). The ‘while’ loop may terminate when a point cj with
positive coordinates is found. That point is a bj -critical point under vj , and
therefore under uj . The point c computed in line 21 is thus an m-critical point
under hλaie , by Corollary A.4(2), and the algorithm returns the correct output.
Alternatively, the ‘while’ loop may terminate when j = 0, and c = c0 is a
b0 -critical point under u0 . But b0 = m and u0 = hλaie , so again the algorithm
returns the correct output.
Algorithm 2 could be improved a bit by using the full strength of Lemma A.3(2)
in the stopping condition for the ‘while’ loop in line 17. Comparing Algorithms 1
and 2, the reader will notice that in Algorithm 2 we are trying to avoid having to
test whether fpt(G) = λ, by performing that test only when p divides m. This
is because that test is often slow when the degree of G is large. If, on the other
hand, a factorization of G is not know from the start, then that test should be
the first thing done in the algorithm.
Algorithm 2 has been implemented by the second author in the Macaylay2
package PosChar [BBH+ ], in the FPT2VarHomog command. To illustrate its use,
below we show a Macaulay2 session that computes the example given in the
introduction:
i1
i2
i3
i4
i5

:
:
:
:
:

installPackage (" PosChar ");
kk = GF ( ZZ /5[ a ]/ ideal ( a ^3+ a +1));
kk [x , y ];
L ={ x ,y , x +y , x + a *y , x + a ^2* y , x + a ^3* y };
FPT2VarHomog (L ,{420 ,419 ,417 ,390 ,402 ,438})
46636216675556057485911762783799675605705641779512143
o5 = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 57968817327716179454988321140262996777892112731933593750
o5 : QQ

If a form in two variable is given, FPT2VarHomog will try to factor the form in
an appropriate extension of the coefficient field and then compute its F -pure
threshold using Algorithm 2:
i6 : ZZ /2[ x , y ];
i7 : G = x ^10* y ^3+ x ^9* y ^4+ x ^6* y ^7+ x ^4* y ^9+ x ^3* y ^10+ x * y ^12+ y ^13;

40

i8 : FPT2VarHomog ( G )
315
o8 = ---2048
o8 : QQ

We invite the reader to download the package and try that command, and
welcome any suggestions or bug reports.

References
[BBH+ ]

E. Bela, D. J. Bruce, D. J. Hernández, Z. Kadyrsizova, M. Katzman, S. Malec,
K. Schwede, P. Teixeira, and E. E. Witt, PosChar, a Macaulay2 package for computations in positive characteristic, available at https://github.com/Macaulay2/
Workshop-2014-Berkeley/tree/master/F-sing.

[BL04]

M. Blickle and R. Lazarsfeld, An informal introduction to multiplier ideals, Trends
in Commutative Algebra, Math. Sci. Res. Inst. Publ., vol. 51, Cambridge Univ.
Press, Cambridge, 2004, pp. 87–114.

[BMS08]

M. Blickle, M. Mustaţă, and K. E. Smith, Discreteness and rationality of F thresholds, Michigan Math. J. 57 (2008), 43–61.

[BMS09]

, F -thresholds of hypersurfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12,
6549–6566.

[BS14]

B. Bhatt and A. K. Singh, The F -pure threshold of a Calabi–Yau hypersurface,
Math. Ann. 362 (2014), no. 1-2, 551–567.

[FST11]

O. Fujino, K. Schwede, and S. Takagi, Supplements to non-lc ideal sheaves, Higher
Dimensional Algebraic Geometry, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011,
pp. 1–46.

[GS]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in
algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.

[Han91]

C. Han, The Hilbert–Kunz function of a diagonal hypersurface, Ph.D. thesis,
Brandeis University, 1991.

[Her11]

D. J. Hernández, F -purity of hypersurfaces, Ph.D. thesis, University of Michigan,
2011, available at http://hdl.handle.net/2027.42/86491.

[Her12]

, F -purity of hypersurfaces, Math. Res. Lett. 19 (2012), no. 2, 389–401.

[HNWZ16] D. J. Hernández, L. Núñez-Betancourt, E. E. Witt, and W. Zhang, F -pure
thresholds of homogeneous polynomials, Michigan Math. J. 65 (2016), 57–87.
[HY03]

N. Hara and K.-i. Yoshida, A generalization of tight closure and multiplier ideals,
Trans. Amer. Math. Soc. 355 (2003), no. 8, 3143–3174.

[Mon06]

P. Monsky, Mason’s theorem and syzygy gaps, J. Algebra 303 (2006), no. 1,
373–381.

[MT04]

P. Monsky and P. Teixeira, p-Fractals and power series—I. Some 2 variable results,
J. Algebra 280 (2004), no. 2, 505–536.

[MT06]

, p-Fractals and power series—II. Some applications to Hilbert–Kunz
theory, J. Algebra 304 (2006), no. 1, 237–255.

[MTW05]

M. Mustaţă, S. Takagi, and K.-i. Watanabe, F-thresholds and Bernstein–Sato
polynomials, European Congress of Mathematics (Zürich), Eur. Math. Soc., 2005,
pp. 341–364.

[Pér13]

F. Pérez, On the constancy regions for mixed test ideals, J. Algebra 396 (2013),
82–97.

41

[Tei02]

P. Teixeira, p-Fractals and Hilbert–Kunz series, Ph.D. thesis, Brandeis University,
2002.

[Tei12]

, Syzygy gap fractals—I. Some structural results and an upper bound, J.
Algebra 350 (2012), no. 1, 132–162.

[TW04]

S. Takagi and K.-i. Watanabe, On F-pure thresholds, J. Algebra 282 (2004), no. 1,
278–297.

42

