Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus\_{n=0}^\infty R\_n$ with $R\_0=k$ (and $R=k[R\_1]$). The Hilbert function $h\_R:\mathbb{N}\rightarrow \mathbb{N}$ is given by $h\_R(n)=\dim\_k R\_n$. When $n\gg 0$, it is known that $h\_R(n)$ agrees with a polnyomial $p\_R(n)$ in $n$ with rational coefficients.


In many theorems in combinatorics, polynomials which count objects have combinatorially sensible meaning when evaluated at negative numbers. For example, the chromatic polynomial $\chi\_G(x)$ of a graph, which counts the number of proper colorings of $G$ with $x$ colors, has the delightful property that $\chi\_G(-1)$ is the number of acyclic orientations of $G$.


One other example, intimately related to the Hilbert function, is that of Ehrhart polynomials. The Ehrhart polynomial $E\_P(n)$ of a convex integral polytope $P$ inside $\mathbb{R}^m$ is the number of lattice points of $\mathbb{Z}^m$ inside the $n$th dilate $nP$. It is also known that $E\_P(-n)$ is the number of interior lattice points (up to sign) of $nP$. For some (all?) polytopes, the Ehrhart polynomial agrees with the Hilbert polynomial of an associated affine semigroup ring.


In general, or in the specific context of affine semigroup rings $R$, is there some sensible (algebraic or combinatorial) meaning to $p\_R(-n)$, or even $p\_R(-1)$?



Hilbert polynomials of graded algebras evaluated at negative numbers
This answer is essentially the same as that of Phil Tosteson, written
before I saw that post. I also mention a non-Cohen-Macaulay example at
the end.


If $R$ is Cohen-Macaulay (but not necessarily generated in degree
one), then $R$ has associated with it a *canonical module*
$\Omega(R)$ which can be graded so its Hilbert function agrees with
$(-1)^d p\_R(-n)$ for $n$ sufficiently large, where $d$ is the Krull
dimension of $R$ or $\Omega(R)$. If $R$ has $n$ generators then it can
be regarded as a module over the polynomial ring
$A=k[x\_1,\dots,x\_n]$. One can then define $\Omega(R) =
\mathrm{Ext}^{n-d}\_A(R,A)$. If $R$ is not Cohen-Macaulay, then there are
"correction terms" to the formula $p\_R(-n) = (-1)^d\mathrm{HQ}(\Omega(R))$,
where $\mathrm{HQ}$ denotes Hilbert quasipolynomial. Namely,
$$ p\_R(-n) = (-1)^d \sum\_{i=0}^d
 (-1)^i\mathrm{HQ}\left( \mathrm{Ext}^{n-d+i}\_A(R,A)\right). $$
(If $R$ is Cohen-Macaulay, then only the term indexed by $i=0$ doesn't
vanish.) I don't know where this result is stated in precisely this
form, but it is equivalent to Theorem 6.4 of my book *Combinatorics
and Commutative Algebra*, second ed. Theorem 8.2 gives an example,
stated in terms of Hilbert series rather than Hilbert quasipolynomials.


