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 a graph.

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 $p_R(-1)$?

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 $p_R(-1)$?