VANDERMONDE VARIETIES AND RELATIONS AMONG
SCHUR POLYNOMIALS

arXiv:1302.1298v1 [math.AG] 6 Feb 2013

RALF FRÖBERG AND BORIS SHAPIRO
Abstract. Motivated by the famous Skolem-Mahler-Lech theorem we initiate
in this paper the study of a natural class of determinantal varieties which we
call Vandermonde varieties. They are closely related to the varieties consisting of all linear recurrence relations of a given order possessing a non-trivial
solution vanishing at a given set of integers. In the regular case, i.e., when the
dimension of a Vandermonde variety is the expected one, we present its free
resolution, obtain its degree and the Hilbert series. Some interesting relations
among Schur polynomials are derived. Many open problems and conjectures
are posed.

1. Introduction
The results in the present paper come from an attempt to understand the famous Skolem-Mahler-Lech theorem and its consequences. Let us briefly recall its
formulation. A linear recurrence relation with constant coefficients of order k is an
equation of the form
un + α1 un−1 + α2 un−2 + · · · + αk un−k = 0, n ≥ k

(1)

where the coefficients (α1 , ..., αk ) are fixed complex numbers and αk 6= 0. (Equation
(1) is often referred to as a linear homogeneous difference equation with constant
coefficients.)
The left-hand side of the equation
tk + α1 tk−1 + α2 tk−2 + · · · + αk = 0

(2)

is called the characteristic polynomial of recurrence (1). Denote the roots of (2)
(listed with possible repetitions) by x1 , . . . , xk and call them the characteristic roots
of (1).
Notice that all xi are non-vanishing since αk 6= 0. To obtain a concrete solution
of (1) one has to prescribe additionally an initial k-tuple, (u0 , . . . , uk−1 ), which can
be chosen arbitrarily. Then un , n ≥ k are obtained by using the relation (1). A
solution of (1) is called non-trivial if not all of its entries vanish. In case of all
distinct characteristic roots a general solution of (1) can be given by
un = c1 xn1 + c2 xn2 + ... + ck xnk
where c1 , ..., ck are arbitrary complex numbers. In the general case of multiple
characteristic roots a similar formula can be found in e.g. [15].
An arbitrary solution of a linear homogeneous difference (or differential) equation with constant coefficients of order k is called an exponential polynomial of
order k. One usually substitutes xi 6= 0 by eγi and considers the obtained function in C instead of Z or N. (Other terms used for exponential polynomials are
quasipolynomials or exponential sums.)
The most fundamental fact about the structure of integer zeros of exponential
polynomials is the well-known Skolem-Mahler-Lech theorem formulated below. It
was first proved for recurrence sequences of algebraic numbers by K. Mahler [11]
2010 Mathematics Subject Classification. Primary 65Q10, Secondary 65Q30, 14M15.
1

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R. FRÖBERG AND B. SHAPIRO

in the 30’s, based upon an idea of T. Skolem [13]. Then, C. Lech [9] published
the result for general recurrence sequences in 1953. In 1956 Mahler published the
same result, apparently independently (but later realized to his chagrin that he had
actually reviewed Lech’s paper some years earlier, but had forgotten it).
Theorem 1 (The Skolem-Mahler-Lech theorem). If a0 , a1 , ... is a solution to a linear recurrence relation, then the set of all k such that ak = 0 is the union of a finite
(possibly empty) set and a finite number (possibly zero) of full arithmetic progressions. (Here, a full arithmetic progression means a set of the form r, r + d, r + 2d, ...
with 0 < r < d.)
A simple criterion guaranteeing the absence of arithmetic progressions is that
no quotient of two distinct characteristic roots of the recurrence relation under
consideration is a root of unity, see e.g. [10]. A recurrence relation (1) satisfying this
condition is called non-degenerate. Substantial literature is devoted to finding the
upper/lower bounds for the maximal number of arithmetic progressions/exceptional
roots among all/non-degenerate linear recurrences of a given order. We give more
details in § 3. Our study is directly inspired by these investigations.
Let Lk be the space of all linear recurrence relations (1) of order at most k with
constant coefficients and denote by L∗k = Lk \ {αk = 0} the subset of all linear
recurrence of order exactly k. (Lk is the affine space with coordinates (α1 , ..., αk ).)
To an arbitrary pair (k; I) where k ≥ 2 is a positive integer and I = {i0 < i1 < i2 <
... < im−1 }, m ≥ k is a sequence of integers, we associate the variety Vk;I ⊂ L∗k , the
set of all linear recurrences of order exactly k having a non-trivial solution vanishing
at all points of I. Denote by Vk;I the (set-theoretic) closure of Vk;I in Lk . We call
Vk;I (resp. V k;I ) the open (resp. closed) linear recurrence variety associated to the
pair (k; I).
In what follows we will always assume that gcd(i1 − i0 , ..., im−1 − i0 ) = 1 to
avoid unnecessary freedom related to the time rescaling in (1). Notice that since
for m ≤ k − 1 one has Vk;I = L∗k and V k;I = Lk , this case does not require special
consideration. A more important observation is that due to translation invariance
of (1) for any integer l and any pair (k; I) the variety Vk;I (resp. V k;I ) coincides
with the variety Vk;I+l (resp. V k;I+l ) where the set of integers I + l is obtained by
adding l to all entries of I.
So far we defined V k;I and Vk;I as sets. However for any pair (k; I) the set
V k;I is an affine algebraic variety, see Proposition 4. Notice that this fact is not
completely obvious since if we, for example, instead of a set of integers choose as I
an arbitrary subset of real or complex numbers then the similar subset of Ln will,
in general, only be analytic.
Now we define the Vandermonde variety associated with a given pair (k; I), I =
{0 ≤ i0 < i1 < i2 < ... < im−1 }, m ≥ k. Firstly, consider the set Mk;I of
(generalized) Vandermonde matrices of the form
 i0

x1
xi20
···
xik0
 xi11
xi21
···
xik1 
,
Mk;I = 
(3)
 ···
···
···
··· 
i

x1m−1

i

x2m−1

···

i

xkm−1

where (x1 , ..., xk ) ∈ Ck . In other words, for a given pair (k; I) we take the map
Mk;I : Ck → M at(m, k) given by (3) where M at(m, k) is the space of all m × kmatrices with complex entries and (x1 , ..., xk ) are chosen coordinates in Ck .
We now define three slightly different but closely related versions of this variety
as follows.

VANDERMONDE VARIETIES AND RELATIONS AMONG SCHUR POLYNOMIALS

3

Version 1. Given a pair (k; I) with |I| ≥ k define the coarse Vandermonde variety
V dck;I ⊂ Mk;I as the set of all degenerate Vandermonde matrices, i.e., whose rank
is smaller than k. V dck;I is obviously an algebraic variety whose defining ideal II

is generated by all m
maximal minors of Mk;I . Denote the quotient ring by
k
RI = R/II .

Denote by Ak ⊂ Ck the standard Coxeter arrangement (of the Coxeter group
Ak−1 ) consisting of all diagonals xi = xj and by BC k ⊂ Ck the Coxeter arrangement
consisting of all xi = xj and xi = 0. Obviously, BCk ⊃ Ak . Notice that V dck;I
always includes the arrangement BC k (some of the hyperplanes with multiplicities)
which is often inconvenient. Namely, with very few exceptions this means that
V dck;I is not equidimensional, not CM, not reduced etc. For applications to linear
recurrences as well as questions in combinatorics and geometry of Schur polynomials
it seems more natural to consider the localizations of V dck;I in Ck \ Ak and in
Ck \ BC k .
c
c
Version 2. Define the Ak -localization V dA
k;I of V dk;I as the contraction of V dk;I
k
A
to C \ Ak . Its is easy to obtain the generating ideal of V dk;I . Namely, recall that
given a sequence J = (j1 < j2 < · · · < jk ) of nonnegative integers one defines the
associated Schur polynomial SJ (x1 , ..., xk ) as given by

xj11
xj2
SJ (x1 , . . . , xk ) = 1
···
xj1k

xj21
xj22
···
xj2k

···
···
···
···

xjk2
xjk2
/W (x1 , . . . , xk ),
···
xjkk

where W (x1 , . . . , xk ) is the usual Vandermonde determinant. Given a sequence
I = (0 ≤ i0 < i1 < i2 < · · · < im−1 ) with gcd(i1 − i0 , . . . , im−1 − i0 ) = 1 consider
the set of all its m
k subsequences Jκ of length k. Here the index κ runs over the
set of all subsequences of length k among {1, 2, ..., m}. Take the corresponding
Schur polynomials SJκ (x1 , . . . , xk ) and form the ideal IIA in the polynomial ring
C[x1 , . . . , xk ] generated by all m
k such Schur polynomials SJκ (x1 , . . . , xk ). One can
k
A
show that the Vandermonde variety V dA
k;I ⊂ C is generated by II , see Lemma 5.
A
A
Denote the quotient ring by RI = R/II where R = C[x1 , ..., xk ]. Analogously, to
the coarse Vandermonde variety V dk;I the variety V dA
k;I often contains irrelevant
coordinate hyperplanes which prevents it from having nice algebraic properties.
For example, if i0 > 0 then all coordinate hyperplanes necessarily belong to V dA
k;I
ruining equidimensionality etc. On the other hand, under the assumption that
i0 = 0 the variety V dA
k;I often has quite reasonable properties presented below.
c
c
Version 3. Define the BC k -localization V dBC
k;I of V dk;I as the contraction of V dk;I
k
BC
to C \ BC k . Again it is straightforward to find the generating ideal of V dk;I .
Namely, given a sequence J = (0 ≤ j1 < j2 < · · · < jk ) of nonnegative integers
define the reduced Schur polynomial ŜJ (x1 , ..., xk ) as given by

1
x1j2 −j1
ŜJ (x1 , . . . , xk ) =
···
x1jk −j1

1
x2j2 −j1
···
x2jk −j1

···
···
···
···

1
xkj2 −j1
/W (x1 , . . . , xk ).
···
jk −j1
xk

In other words, ŜJ (x1 , ..., xk ) is the usual Schur polynomial corresponding to the
sequence (0, j2 − j1 , ...., jk − j1 ). Given a sequence I = (0 ≤ i0 < i1 < i2 < · · · <
im−1 ) with gcd(i1 − i0 , . . . , im−1 − i0 ) = 1 consider as before the set of all its m
k
subsequences Jκ of length k where the index κ runs over the set of all subsequences
of length k. Take the corresponding reduced Schur polynomials ŜJκ (x1 , . . . , xk )

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R. FRÖBERG AND B. SHAPIRO


and form the ideal IIBC in the polynomial ring C[x1 , . . . , xk ] generated by all m
k
such Schur polynomials ŜJκ (x1 , . . . , xk ). One can easily see that the Vandermonde
k
BC
BC
variety V dBC
=
k;I ⊂ C as generated by II . Denote the quotient ring by RI
BC
R/II .
BC
Conjecture 2. If dim(V dBC
is a radical ideal.
k;I ) ≥ 2 then II

Notice that considered as sets the restrictions to Ck \ BC k of all three varieties
BC
V dck;I , V dA
k;I , V dk;I coincide with what we call the open Vandermonde variety
op
V dk;I which is the subset of all matrices of the form Mk;I with three properties:
(i) rank is smaller than k;
(ii) all xi ’s are non-vanishing;
(iii) all xi ’s are pairwise distinct.
Thus set-theoretically all the differences between the three Vandermonde varieties are concentrated on the hyperplane arrangement BC k . Also from the above
BC
definitions it is obvious that V dop
k;I and V dk;I are invariant under addition of an
arbitrary integer to I. The relation between the linear recurrence variety Vk;I and
the open Vandermonde variety V dop
k;I is quite straight-forward. Namely, consider
the standard Vieta map:
V i : Ck → L k
(4)
sending an arbitrary k-tuple (x1 , ..., xk ) to the polynomial tk +α1 tk−1 +α2 tk−2 +· · ·+
αk whose roots are x1 , ...., xk . Inverse images of the Vieta map are exactly the orbits
of the standard Sk -action on Ck by permutations of coordinates. Thus, the Vieta
map sends a homogeneous and symmetric polynomial to a weighted homogeneous
polynomial.
op
Define the open linear recurrence variety Vk;I
⊆ Vk;I of a pair (k; I) as consisting
of all recurrences in Vk;I with all distinct characteristic roots distinct. The following
statement is obvious.
Lemma 3. The map V i restricted to V dop
k;I gives an unramified k!-covering of the
op
set Vk;I
.
Unfortunately at the present moment the following natural question is still open.
op
op
= Vk;I where Vk;I
Problem 1. Is it true that for any pair (k; I) one has that Vk;I
op
is the set-theoretic closure of Vk;I in L∗k ? If ’not’, then under what additional
assumptions?

Our main results are as follows. Using the Eagon-Northcott resolution of determinantal ideals, we determine the resolution, and hence the Hilbert series and
degree of RA
I in Theorem 6. We give an alternative calculation of the degree using
the Giambelli-Thom-Porteous formula in Proposition 8. In the simplest non-trivial
case, when m = k+1, we get more detailed information about V dA
k;I . We prove that
its codimension is 2, and that RA
is
Cohen-Macaulay.
We
also
discuss
minimal sets
I
of generators of II , and determine when we have a complete intersection in Theorem 9. (The proof of this theorem gives a lot of interesting relations between Schur
polynomials, see Theorem 10.) In this case the variety has the expected codimension, which is not always the case if m > k + 1. In fact our computer experiments
suggest that then the codimension rather seldom is the expected one. In case k = 3,
m = 5, we show that having the expected codimension is equivalent to RA
I being a
complete intersection and that II is generated by three complete symmetric functions. Exactly the problem (along with many other similar questions) when three
complete symmetric functions constitute a regular sequence was considered in an

VANDERMONDE VARIETIES AND RELATIONS AMONG SCHUR POLYNOMIALS

5

recent paper [4] where the authors formulated a detailed conjecture. We slightly
strengthen their conjecture below.
For the BC k -localized variety V dBC
k;I we have only proofs when k = 3, but we
present Conjectures 15 and 16, supported by many calculations. We end the paper
with a section which describes the connection of our work with the fundamental
problems in linear recurrence relations.
Acknowledgements. The authors want to thank Professor Maxim Kazarian (Steklov
Institute of Mathematical Sciences) for his help with Giambelli-Thom-Porteous formula, Professor Igor Shparlinski (Macquarie University) for highly relevant information on the Skolem-Mahler-Lech theorem and Professors Nicolai Vorobjov (University of Bath) and Michael Shapiro (Michigan State University) for discussions.
We are especially grateful to Professor Winfried Bruns (University of Osnabrück)
for pointing out important information on determinantal ideals.
2. Results and conjectures on Vandermonde varieties
We start by proving that V k;I is an affine algebraic variety, see Introduction.
Proposition 4. For any pair (k; I) the set V k;I is an affine algebraic variety.
Therefore, Vk;I = V k;I |L∗k is a quasi-affine variety.
Proof. We will show that for any pair (k; I) the variety V k;I of linear recurrences
is constructible. Since it is by definition closed in the usual topology of Lk ≃ Ck
it is algebraic. The latter fact follows from [12], I.10 Corollary 1 claiming that
if Z ⊂ X is a constructible subset of a variety, then the Zarisky closure and the
strong closure of Z are the same. Instead of showing that V k;I is constructible we
prove that Vk;I ⊂ L∗k is constructible. Namely, we can use an analog of Lemma 3 to
construct a natural stratification of Vk;I into the images of quasi-affine
sets under
S
λ
appropriate Vieta maps. Namely, let us stratify Vk;I as Vk;I = λ⊢k Vk;I
where
λ
λ ⊢ k is an arbitrary partition of k and Vk;I is the subset of Vk;I consisting of all
recurrence relations of length exactly k which has a non-trivial solution vanishing
at each point of I and whose characteristic polynomial determines
the partition
Ps
λ of its degree k. In other words, if λ = (λ1 , ..., λs ),
λ
=
k then the
j
j=1
characteristic polynomial should have s distinct roots of multiplicities λ1 , ..., λs
λ
resp. Notice that any of these Vk;I
can be empty including the whole Vk;I in which
λ
case there is nothing to prove. Let us now show that each Vk;I
is the image under
the appropriate Vieta map
of
a
set
similar
to
the
open
Vandermonde
variety. Recall
Ps
that if λ = (λ1 , ..., λs ),
λ
=
k
and
x
,
..,
x
are
the
distinct
roots
with the
1
s
j=1 j
multiplicities λ1 , ..., λs respectively of the linear recurrence (1) then the general
solution of (1) has the form
un = Pλ1 (n)xn1 + Pλ2 (n)xn2 + ... + Pλs (n)xns ,
where Pλ1 (n), ..., Pλs (n) are arbitrary polynomials in the variable n of degrees λ1 −
1, λ2 − 1, ...., λs−1 resp. Now, for a given λ ⊢ k consider the set of matrices
λ
Mk;I
=

xi10
 xi1
 1
 ···
i
x1m−1


i0 xi10
i1 xi11
···
i
im−1 x1m−1

...
...
···
...

i0λ1 −1 xi10
i1λ1 −1 xi11
···
λ1 −1 im−1
im−1
x1

...
...
···
...

xis0
xis1
···
i
xsm−1

i0 xis0
i1 xis1
···
i
i1 xsm−1

...
...
···
...


i0λs −1 xis0
i1λs −1 xis1 
.

···
λs −1 im−1
im−1 xs

In other words, we are taking the fundamental solution xn1 , nxn1 , ..., nλ1 −1 xn1 , xn2 ,
nxn2 , ..., nλ2 −1 xn2 , ...., xns , nxns , ..., nλs −1 xns of (1) under the assumption that the
characteristic polynomial gives a partition λ of k and we are evaluating each function

6

R. FRÖBERG AND B. SHAPIRO

in this system at i0 , i1 , ..., im−1 resp. We now define the variety V dλk;I as the subset
λ
of matrices of the form Mk;I
such that: (i) the rank of such a matrix is smaller
than k; (ii) all xi are distinct; (iii) all xi are non-vanishing. Obviously, V dλk;I is a
quasi-projective variety in Cs . Define the analog
V iλ : Cs → Lk which sends an
Qs
s
s-tuple (x1 , ..., xs ) ∈ C to the polynomials j=1 (x − xj )λj ∈ Lk of the Vieta map
λ
V i. One can easily see that V iλ maps V dλk;I onto Vk;I
. Applying this construction
S
λ
is constructible which
to all partitions λ ⊢ k we will obtain that Vk;I = λ⊢k Vk;I
finishes the proof.

The remaining part of the paper is devoted to the study of the Vandermonde
BC
A
varieties V dA
k;I and V dk;I . We start with the Ak -localized variety V dk;I . Notice
A
k
that if m = k the variety V dk;I ⊂ C is an irreducible hypersurface given by the

Pk−1
equation SI = 0 and its degree equals j=0
ij − k2 . We will need the following
alternative description of the ideal IIA in the general case. Namely, using the
Jacobi-Trudi identity for the Schur polynomials we get the following statement.
Lemma 5. For any pair (k; I), I = {i0 < i1 < ... < im−1 } the ideal IIA is generated
by all k × k-minors of the m × k-matrix




Hk;I = 


hi0 −(k−1)
hi1 −(k−1)
..
.

hi0 −(k−2)
hi1 −(k−2)
..
.

···
···
..
.

hi0
hi1
..
.

him−1 −(k−1)

him−1 −(k−2)

···

him−1





.


(5)

Here hi denotes the complete symmetric function of degree i, hi = 0 if i < 0,
h0 = 1.
Proof. It follows directly from the standard Jacobi-Trudi identity for the Schur
polynomials, see e.g. [16].

In particular, Lemma 5 shows that V dA
k;I is an determinantal variety in the usual
A
sense. When working with V dk;I and unless the opposite is explicitly mentioned we
will assume that I = {0 < i1 < ... < im−1 }, i.e. that i0 = 0 and that additionally
gcd(i1 , ..., im−1 ) = 1. Let us first study some properties of V dA
k;I in the so-called
regular case, i.e. when its dimension coincides with the expected one.
Namely, consider the set Ωm,k ⊂ M at(m, k) of all m× k-matrices having positive
corank. It is well-known that Ωm,k has codimension equal to m − k + 1. Since V dck;I
coincides with the pullback of Ωm,k under the map Mk;I and V dA
k;I is closely related
to it (but with trivial pathology on Ak removed) the expected codimension of V dA
k;I
equals m − k + 1. We call a pair (k; I) A-regular if k ≤ m ≤ 2k − 1 (implying that
A
the expected dimension of V dA
k;I is positive) and the actual codimension of V dk;I
coincides with its expected codimension. We now describe the Hilbert series of the
quotient ring RA
I in the case of a arbitrary regular pair (k; I) using the well-known
resolution of determinantal ideals of Eagon-Northcott [6].
To explain the notation in the following theorem, we introduce two gradings,
tdeg and deg, on C[t0 , . . . , tm−1 ]. The first one is the usual grading induced by
tdeg(ti ) = 1 for all i, and a second one is induced by deg(ti ) = −i. In the next
theorem M denotes a monomial in C[t0 . . . , tm−1 ].
Theorem 6. In the above notation, and with I ′ = {i1 , . . . , im−1 },

VANDERMONDE VARIETIES AND RELATIONS AMONG SCHUR POLYNOMIALS

7

A
A
(a) the Hilbert series HilbA
I (t) of RI = R/II is given by
P
Pm−k
P
1 − i=0 ((−1)i+1 J⊆I ′ ,|J|=k+i tsJ M∈Ni tdeg(M) )
A
HilbI (t) =
,
(1 − t)k

P
where sJ = j∈J j − k2 and Ni = {M ; tdeg(M ) = i}.

(m−k+1)
(b) The degree of RA
(1)(−1)m−k+1 /(m − k + 1)!, where T (t) is
I is T
the numerator in (a).

Proof. According to [6] provided that IIA has the expected codimension m − k + 1,
it is known to be Cohen-Macaulay and it has a resolution of the form
0 → Fm−k+1 → · · · → F1 → R → RA
(6)
I → 0,


m
k+j−2
where Fj is free module over R = C[x1 , . . . , xk ] of rank k+j−1
k−1 . We denote
the basis elements of Fj by MI T , where I ⊆ {i0 , . . . .im−1 }, |I| = k + j − 1, and T is
an arbitrary monomial in {t0 , . . . , tk−1 } of degree j − 1. If MI = {il1 , . . . , ilk+j−1 }
Pr
and T = tsj11 · · · tsjrr with si > 0 for all i and i=1 si = j − 1, then, in our sitPr Pk+j−1
uation, d(MI T ) = i=1 ( l=1 (−1)k+1 hikl −ji MI\{ikl } )T /tji . Here deg(MI ) =

Pk+j−1
ilj − k2 and deg(ti ) = −i. (Note that tdeg(ti ) = 1 but deg(ti ) = −i.)
n=1

Pr
Pk+j−1
Thus deg(MI T ) =
iln + i=1 si ji − k2 if MI = {il1 , . . . , ilk+j−1 } and
n=1
T = tsj11 · · · tsjrr . Observe that this resolution is never minimal. Indeed, for any sequence I = {0 = i0 < i1 · · · < im−1 }, we only need the Schur polynomials coming
from subsequences starting with 0, so IIA is generated by at most m−1
Schur
k−1

polynomials instead of totally m
;
see
also
discussions
preceding
the
proof
of
Thek
orem 9 below. Now, if J is an arbitrary homogeneous ideal in R = C[x1 , . . . , xk ]
and R/J has a resolution
1
r
R(−n1.i ) → R → R/J → 0,
R(−nr,i ) → · · · → ⊕βi=1
0 → ⊕βi=1

then the Hilbert series of R/J is given by
P 1 n1,i
P r nr,i
1 − βi=1
+ · · · + (−1)r βi=1
t
t
.
k
(1 − t)
For the resolution (6), all terms coming from MI T with t0 |T cancel. Thus we
get the claimed Hilbert series. If the Hilbert series is given by T (t)/(1 − t)k =
P (t)/((1 − t)dim(R/II ) , then the degree of the corresponding variety equals P (1).
We have T (t) = (1−t)m−k+1 P (t), or after differentiating the latter identity m−k+1
times we get P (1) = T (m−k+1) (1)(−1)m−k+1 /(m − k + 1)!.

Example 7. For the case 3 × 5 with I = {0, i1 , i2 , i3 , i4 ) , if the ideal V dA
k;I has
the right codimension, we get that its Hilbert series equals T (t)/(1 − t)3 , where
T (t) = 1−ti1 +i2 −2 −ti1 +i3 −2 −ti1 +i4 −2 −ti2 +i3 −2 −ti2 +i4 −2 −ti3 +i4 −2 +ti1 +i2 +i3 −3 +
ti1 +i2 +i4 −3 + ti1 +i3 +i4 −3 + ti2 +i3 +i4 −3 + ti1 +i2 +i3 −4 + ti1 +i2 +i4 −4 + ti1 +i3 +i4 −4 +
ti2 +i3 +i4 −4 − ti1 +i2 +i3 +i4 −3 − ti1 +i2 +i3 +i4 −4 − ti1 +i2 +i3 +i4 −5
and the degree of V dA
k;I equals
i1 i2 i3 +i1 i2 i4 +i1 i3 i4 +i2 i3 i4 −3(i1 i2 +i1 i3 +i1 i4 +i2 i3 +i2 i4 +i3 i4 )+7(i1 +i2 +i3 +i4 )−15.
An alternative way to calculate deg(V dA
k;I ) is to use the Giambelli-Thom-Porteous
formula, see e.g. [8]. The next result corresponded to the authors by M. Kazarian
explains how to do that.

8

R. FRÖBERG AND B. SHAPIRO

Proposition 8. Assume that V dA
k;I has the expected codimension m − k + 1. Then
its degree (taking multiplicities of the components into account) is equal to the
coefficient of tm−k+1 in the Taylor expansion of the series
Qm−1
j=1 (1 + ij t)
.
Qk−1
j=1 (1 + jt)
More explicitly,

deg(V dA
k;I ) =

m−k+1
X

σj (I)um−k+1−j

j

where σj is the jth elementary symmetric function of the entries (i1 , ..., im−1 ) and
Qk−1 1
u0 , u1 , u2 , ... are the coefficients in the Taylor expansion of j=1 1+jt
, i.e. u0 +

 3k−2
Qk−1 1
u1 t + u2 t2 + ... = j=1 1+jt
. In particular, u0 = 1, u1 = − k2 , u2 = k+1
4 ,
3
 

k+2 k
k+3 15k3 −15k2 −10k+8
u3 = − 4 2 , u4 = 5
.
48

Proof. In the Giambelli formula setting, we consider a ”generic” family of (n × l)matrices A = ||ap,q ||, 1 ≤ p ≤ n, 1 ≤ q ≤ l, whose entries are homogeneous
functions of degrees deg(ap,q ) = αp − βq in parameters (x1 , ..., xk ) for some fixed
sequences β = (β1 , ..., βl ) and α = (α1 , ..., αn ). Denote by Σr the subvariety in the
parameter space Ck determined by the condition that the matrix A has rank at
most l − r, that is, the linear operator A : Cl → Cn has at least a r-dimensional
kernel. Then the expected codimension of the subvariety Σr is equal to
codim(Σr ) = r(n − l + r).
In case when the actual codimension coincides with the expected one its degree is
computed as the following r × r-determinant:
deg(Σr ) = det ||cn−l+r−i+j ||1≤i, j≤r ,

(7)

where the entries ci ’s are defined by the Taylor expansion
Qn
p=1 (1 + αp t)
2
.
1 + c1 t + c2 t + ... = Ql
q=1 (1 + βq t)

There is a number of situations where this formula can be applied. Depending
on the setting, the entries αp , βq can be rational numbers, formal variables, first
Chern classes of line bundles or formal Chern roots of vector bundles of ranks n
and l, respectively. In the situation of Theorem 6 we should use the presentation
(5) of V dA
k;I from Lemma 5. Then we have n = m, l = k, r = 1, α = I =
(0, i1 , ..., im−1 ), β = (k − 1, k − 2, ..., 0). Under the assumptions of Theorem 6 the
degree of the Vandermonde variety V dA
k;I will be given by the 1 × 1-determinant of
the Giambelli-Thom-Porteous formula (7), that is, the coefficient cm−k+1 of tm−k+1
in the expansion of
Qm−1
Qm−1
j=1 (1 + ij t)
j=0 (1 + ij t)
2
= Qk−1
,
1 + c1 t + c2 t + ... = Qk
j=1 (1 + (k − j)t)
j=1 (1 + jt)
which gives exactly the stated formula for deg(V dA
k;I ).



In the simplest non-trivial case m = k + 1 one can obtain more detailed information about V dA
k;I . Notice that for m = k + 1 the k + 1 Schur polynomials
generating the ideal IIA are naturally ordered according to their degree. Namely,
given an arbitrary I = {0 < i1 < i1 < ... < ik } with gcd(i1 , ..., ik ) = 1 denote by
Sj , j = 0, ..., k the Schur polynomial obtained by removal of the (j)-th row of the
matrix Mk;I . (Pay attention that here we enumerate the rows starting from 0.)

VANDERMONDE VARIETIES AND RELATIONS AMONG SCHUR POLYNOMIALS

9

Then, obviously, deg Sk < deg Sk−1 < .... < deg S0 . Using presentation (5) we get
the following.
Theorem 9. For any integer sequence I = {0 = i0 < i1 < i2 < ... < ik } of length
k + 1 with gcd(i1 , ..., ik ) = 1 the following facts are valid:
(i) codim(V dA
k;I ) = 2;
(ii) The quotient ring RA
I is Cohen-Macaulay;
A
(iii) The Hilbert series HilbA
I (t) of RI is given by the formula


k−1
k
X
X
k
k
1 −
tN −j−( 2)  /(1 − t)k ,
tN −ij −( 2) +
HilbA
I (t) =
j=1

where N =

Pk

j=1
P

j=1

ij ;

 Pk

k+1
(iv) deg(V
= 1≤j<l≤k ij il − k2
j=1 ij +
3 (3k − 2)/4;
(v) The ideal IIA is always generated by k generators Sk , ..., S1 (i.e., the last
generator S0 always lies in the ideal generated by Sk , ..., S1 ). Moreover,
if for some 1 ≤ n ≤ k − 2 one has in ≤ k − n then IIA is generated by
k − n elements Sk , ..., Sn+1 . In particular, it is generated by two elements
Sk , Sk−1 (i.e., is a complete intersection) if ik−2 ≤ k − 1.
dA
k;I )

The theorem gives lots of relations between Schur polynomials.
Theorem 10. Let the generators be Sk = sik−1 −k+1,ik−2 −k+2,...,i1 −1 , Sk−1 , . . . , S0 =
sik −(k−1),ik−1 −(k−2),...,i1 in degree increasing order. For s = 0, 1 . . . , k − 1 we have
hik −s Sk − hik−1 −s Sk−1 + · · · + (−1)k−1 hi1 −s S1 + (−1)k h−s S0 = 0.
Here hi = 0 if i < 0.
To prove Theorems 9 and 10 notice that since Schur polynomials are irreducible
[5], in the case m = k + 1 the ideal IIA always has the expected codimension 2
unless it coincides with the whole ring C[x1 , ..., xk ]. Therefore vanishing of any
two Schur polynomials lowers the dimension by two. (Recall that we assume that
gcd(i1 , ..., ik ) = 1.) On the other hand, as we mentioned in the introduction the
codimension of V dA
k;I in this case is at most 2. For m = k + 1 one can present a
very concrete resolution of the quotient ring RA
I .
Namely, given a sequence I = {0 = i0 < i1 < · · · < ik } we know that the ideal
IIA is generated by the k + 1 Schur polynomials Sl = sak ,ak−1 ,...,a1 , l = 0, . . . , k,
where
(ak , . . . , a1 ) = (ik , ik−1 , . . . , il+1 , iˆl , il−1 , . . . , i0 ) − (k − 1, k − 2, . . . , 1, 0).

P
Obviously, Sl has degree kj=1 ij − il − k2 and by the Jacobi-Trudi identity is given
by
hi0 −(k−1)
hi0 −(k−2) · · ·
hi0
hi1 −(k−1)
hi1 −(k−2) · · ·
hi1
..
..
..
..
.
.
.
.
Sl =

hil−1 −(k−1)
hil+1 −(k−1)
..
.

hil−1 −(k−2)
hil+1 −(k−2)
..
.

···
···
..
.

hil−1
hil+1
..
.

hik−1 −(k−1)
hik −(k−1)

hik−1 −(k−2
hik −(k−2)

···
···

hik−1
hik

.

Here (as above) hj denotes the complete symmetric function of degree j in x1 , ..., xk .
(We set hj = 0 if j < 0 and h0 = 1.) Consider the (k + 1) × k-matrix H = Hk;I

10

R. FRÖBERG AND B. SHAPIRO

given by


hi0 −(k−1)
hi1 −(k−1)
..
.

hi0 −(k−2)
hi1 −(k−2)
..
.

···
···
..
.

hi0
hi1
..
.









H=
.


 hi −(k−1) hi −(k−2 · · · hik−1 
k−1
k−1
hik −(k−1)
hik −(k−2) · · ·
hik
Let Hl be the (k + 1) × (k + 1)-matrix obtained by extending H with l-th column
of H. Notice that det(Hl ) = 0, and expanding it along the last column we get for
0 ≤ l ≤ k − 1 the relation
0 = det(Hl ) = hik −(k−l) Sk − hik−1 −(k−l) Sk−1 + · · · + (−1)k−1 hi1 −(k−l) S1 .
For l = k we get
hik Sk − hik−i Sk−1 + · · · + (−1)k hi0 S0 = 0
which implies that S0 always lie in the ideal generated by the remaining S1 , . . . , Sk .
We now prove Theorem 9.
P
Proof. Set N = kj=1 ij . For an arbitrary I = {0, i1 , ..., ik ) with gcd(i1 , ..., ik ) = 1
A
we get the following resolution of the quotient ring RA
I = R/II
 
 
k
k
k−1
0 −→ ⊕kl=1 R(−N +
+ l) −→ ⊕l=1
R(−N + il +
) −→ R −→ RA
I −→ 0
2
2

where R = C[x1 , ..., xk ]. Simple calculation with this resolution implies that the
A
Hilbert series HilbA
I (t) of RI is given by
!
k−1
k
X
X
k
k
tN −(2)−l /(1 − t)k
HilbA (t) = 1 −
tN −il −(2 ) +
I

l=1

l=1

and the degree of V dA
k;I is given by
deg(V dA
k;I ) =

X

1≤r<s≤k

ir is −

 X


k
k
k+1
ir +
(3k − 2)/4.
2 r=1
3

Notice that the latter resolution might not be minimal, since the ideal might have
fewer than k generators. To finish proving Theorem 9 notice that if conditions of (v)
are satisfied then a closer look at the resolution reveals that the Schur polynomials
S0 , . . . , Sk−n lie in the ideal generated by Sk−n+1 , . . . , Sk .

In connection with Theorems 6 and 9 the following question is completely natural.
Problem 2. Under the assumptions i0 = 0 and gcd(i1 , ..., im−1 ) = 1 which pairs
(k; I) are A-regular?
Theorem 9 shows that for m = k + 1 the condition gcd(i1 , ..., ik ) = 1 guarantees
regularity of any pair (k; I) with |I| = k + 1. On the other hand, our computer
experiments with Macaulay suggest that for m > k regular cases are rather seldom.
In particular, we were able to prove the following.
Theorem 11. If m > k a necessary (but insufficient) condition for V dA
k;I to have
the expected codimension is i1 = 1.
Proof. If i1 ≥ 2, then ik−2 ≥ k − 1. This means that the ideal is generated by
Schur polynomials sa0 ,...,ak−1 with ak−2 ≥ 1. Multiplying these up to degree n
gives linear combinations of Schur polynomials sb1 ,...,bk−1 with bk−2 ≥ 1. Thus we
miss all Schur polynomials with bk−2 = 0. The number of such Schur polynomials

VANDERMONDE VARIETIES AND RELATIONS AMONG SCHUR POLYNOMIALS

11

equals the number of partitions of n in at most k−2 parts. The number of partitions
of n in exactly k − 2 parts is approximated with nk−3 /((k − 2)!(k − 1)!). Thus the
number of elements of degree n in the ring is at least cnk−3 for some positive c, so
the ring has dimension ≥ k − 2. The expected dimension is ≤ k − 3, which is a
contradiction.

So far a complete (conjectural) answer to Problem 2 is only available in the
first non-trivial case k = 3, m = 5. Namely, for a 5-tuple I = {0, 1, i2, i3 , i4 } to
be regular one needs the corresponding the Vandermonde variety V dA
3;I to be a
complete intersection. This is due to the fact that in this situation the ideal IIA
is generated by the Schur polynomials S4 , S3 , S2 of the least degrees in the above
notation. Notice that S4 = hi2 −2 , S3 = hi3 −2 , S2 = hi4 −2 . Thus V dA
3;I has the
expected codimension (equal to 3) if and only if C[x1 , x2 , x3 ]/hhi2 −2 , hi3 −2 , hi4 −2 i is
a complete intersection or, in other words, hi2 −2 , hi3 −2 , hi4 −2 is a regular sequence.
Exactly this problem (along with many other similar questions) was considered in
intriguing paper [4] where the authors formulated the following claim, see Conjecture 2.17 of [4].
Conjecture 12. Let A = {a, b, c} with a < b < c. Then ha , hb , hc in three variables
is a regular sequence if and only if the following conditions are satisfied:
(1) abc ≡ 0 mod 6;
(2) gcd(a + 1, b + 1, c + 1) = 1;
(3) For all t ∈ N with t > 2 there exists d ∈ A such that d + 2 6≡ 0, 1 mod t.
In fact, our experiments allow us to strengthen the latter conjecture in the following way.
Conjecture 13. In the above set-up if the sequence ha , hb , hc with a > 1 in three
variables is not regular, then hc lies in the ideal generated by ha and hb . (If
(a, b, c) = (1, 4, 3k + 2), k ≥ 1, then ha , hb , hc neither is a regular sequence, nor
hc ∈ (ha , hb ).)
We note that if we extend the set-up of [4] by allowing Schur polynomials s(r, s, t)
instead of just complete symmetric functions then if t > 0 in all three of them the
sequence is never regular. Conjectures 12 and 13 provide a criterion which agrees
with our calculations of dim(V dA
3;I ). Finally, we made experiments checking how
)
depends
on
the
last
entry
im−1 of I = {0, 1, i2, ..., im−1 } while keeping
dim(V dA
k;I
the first m − 1 entries fixed.
Conjecture 14. For any given I = (0, 1, i2 , ..., im−1 ) the dimension dim(V dA
k;I )
depends periodically on im−1 for all im−1 sufficiently large.
Notice that Conjecture 14 follows from Conjecture 12 in the special case k =
3, m = 5. Unfortunately, we do not have a complete description of the length of
this period in terms of the fixed part of I and it might be quite tricky.
For the BC k -localized variety V dBC
k;I we have, except for k = 3, only conjectures,
supported by many calculations.
Conjecture 15. For any integer sequence I = {0 = i0 < i1 < i2 < ... < ik } of
length k + 1 with gcd(i1 , ..., ik ) = 1 the following facts are valid.
(i) codim(V dBC
k;I ) = 2;
is Cohen-Macaulay;
(ii) The quotient ring RBC
I
of the form
(iii) There is a C[x1 , . . . , xn ] = R-resolution of RBC
I
 
 
k
k
k−1
0 → ⊕j=0
R[−N +j+
] → ⊕kj=1 R[−N +ij +
]⊕R[−N +ki1] → R → R → RBC
→0
I
2
2

12

R. FRÖBERG AND B. SHAPIRO
BC
(iv) The Hilbert series HilbBC
is given by the formula
I (t) of RI


 
k−1
k
X
X
k
k 
1 −
tN −i1 −j −
/(1 − t)k
tN −j−( 2) − tN −ki1 +
HilbBC
I (t) =
2
j=1
j=1

where N =

Pk

j=1 ij ;

 Pk


P
k+1
k
(v) deg(V
= 1≤j<l≤k ij il − k2
j=1 ij +
3 (3k − 2)/4 − 2 i1 (i1 − 1);
(vi) The ideal IIBC is always generated by k generators. It is generated by two
elements (i.e., is a complete intersection) if i1 ≤ k − 1.
dBC
k;I )

Conjecture 16. Let Sk , . . . , S1 be as in Theorem 6 and G0 = sik −i1 −k+1,...,i2 −i1 −1 .
Then, for s = 0, . . . , k − 1 we have
hik −i1 −s Sk − hik−1 −i1 −s Sk−1 + · · · + (−1)k−2 hi2 −i1 −s S2 + (−1)k−1 h−s S1 +
(−1)k si1 −1,...,(i1 −1)k−1 ,k−1−s G0 .
Here hi = 0 if i < 0 and hi,...,i,j = 0 if j > i, and (i1 − 1)k−1 means i1 − 1, . . . , i1 − 1
(k − 1 times).
That the ring is CM follows from the fact that the ideal is generated by the
maximal minors of a t × m-matrix in the ring of Laurent polynomials. To prove
the theorem it suffices to prove the relations between the Schur polynomials. Unfortunately we have managed to do that only for k = 3.
3. Final remarks
Here we briefly explain the source of our interest in Vandermonde varieties. In
1977 J. H. Loxton and A. J. van der Poorten formulated an important conjecture
(Conjecture 1′ of [10]) claiming that there exists a constant µk such that any integer
recurrence of order k either has at most µk integer zeros or has infinitely many zeros.
This conjecture was first settled by W. M. Schmidt in 1999, see [14] and also by
J. H. Evertse and H. P. Schlickewei, see [7].
The upper bound for µk obtained in [14] was
e3k log k

µk < e e

,

which was later improved by the same author to
e20k

µk < e e

.

Apparently the best known at the moment upper bound for µk was obtained in
[1] and is given by
√
k

11k

µk < e e
.
Although the known upper bounds are at least double exponential it seems
plausible that the realistic upper bounds should be polynomial. The only known
nontrivial lower bound for µk was found in [2] and is given by


k+1
µk ≥
− 1.
2

One should also mention the non-trivial exact result of F. Beukers showing that for
sequences of rational numbers obtained from recurrence relations of length 3 one
has µ3 = 6, see [3].
The initial idea of this project was to try to obtain upper/lower bounds for
µk by studying algebraic and geometric properties of Vandermonde varieties but
they seems to be quite complicated. Let us finish with some further problems and
comments on them, that we got with an extensive computer search. Many questions

VANDERMONDE VARIETIES AND RELATIONS AMONG SCHUR POLYNOMIALS

13

related to the Skolem-Mahler-Lech theorem translate immediately into questions
about Vk;I . For example, one can name the following formidable challenges.
Problem 3. For which pairs (k; I) the variety Vk;I is empty/non-empty? More
generally, what is the dimension of Vk;I ?
We made a complete computer search for RA
I and some variants where we removed solutions on the coordinate planes and axes, and looked for arithmetic sequences, for (0, i1 , i2 , i3 ), 0 < i1 < i2 < i3 , i3 ≤ 13 (so k = 3, m = 4). The only
cases when Vk;I was empty were I = (0, 1, 3, 7) and I = (0, 1, 3, 9) and their ”duals”
(0, 4, 6, 7) and (0, 6, 8, 9). We suspect that our exceptions are the only possible. For
k = 3, m = 5 we investigated I = (0, i1 , i2 , i3 , i4 ), 0 < i1 < i2 < i3 < i4 , i4 ≤ 9.
For i1 = 1 about half of the cases had the expected dimension. For (k, m) = (3, 6),
i5 ≤ 10, for (k, m) = (4, 6), i5 ≤ 9 and for (k, m) = (5, 8), i7 ≤ 10, most cases were
of expected dimension. The corresponding calculations for RBC
I , (k, m) = (3, 5),
i4 ≤ 9, showed that about half of the cases had expected codimension.
Problem 4. For which pairs (k; I) any solution of a linear recurrence vanishing at I
must have an additional integer root outside I? More specifically, for which pairs
(k; I) any solution of a linear recurrence vanishing at I must vanish infinitely many
times in Z? In other words, for which pairs (k; I) the set of all integer zeros of
the corresponding solution of any recurrence relation from Vk;I must necessarily
contain an arithmetic progression?
For example, in case k = 3, m = 4 we found that the first situation occurs
for 4-tuples (0, 1, 4, 6) and (0, 1, 4, 13) which both force a non-trivial solution of a
third order recurrence vanishing at them to vanish at the 6-tuple (0, 1, 4, 6, 13, 52),
which is the basic example in [3]. The second situation occurs if in a 4-tuple I =
{0, i1 , i2 , i3 } two differences between its entries coincide, see [3]. But this condition
is only sufficient and no systematic information is available. Notice that for any
pair (k; I) the variety V k;I is weighted-homogeneous where the coordinate αi , i =
1, . . . , k has weight i. (This action corresponds to the scaling of the characteristic
roots of (2).)
We looked for cases containing an arithmetic sequence with difference at most
10 and we found cases which gave arithmetic sequences with difference 2,3,4, and
5, and a few cases which didn’t give any arithmetic sequences.
Problem 5. Is it true that if an (k + 1)-tuple I consists of two pieces of arithmetic
progression with the same difference then any exponential polynomial vanishing at
I contains an arithmetic progression of integer zeros?
Problem 6. If the answer to the previous question is positive is it true that there are
only finitely many exceptions from this rule leading to only arithmetic progressions?
Finally a problem similar to that of J. H. Loxton and A. J. van der Poorten can
be formulated for real zeros of exponential polynomials instead of integer. Namely,
the following simple lemma is true.
Lemma 17. Let λ1 , ..., λn be a arbitrary finite set of (complex) exponents having
all distinct real parts then an arbitrary exponential polynomial of the form c1 eλ1 z +
c2 eλ2 z + .. + cn eλn z , ci ∈ C has at most finitely many real zeros.
Problem 7. Does there exist an upper bound on the maximal number real for the
set of exponential polynomials given in the latter lemma in terms of n only?
Problem 8. What about non-regular cases? Describe their relation to the existence
of additional integer zeros and arithmetic progressions as well as additional Schur
polynomials in the ideals.

14

R. FRÖBERG AND B. SHAPIRO

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Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
E-mail address: ralff@math.su.se
Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
E-mail address: shapiro@math.su.se

