Multivariate quasipolynomials and where to find them

Question

This question is inspired from thinking about David Speyer's question about complex variable Ehrhart theory.

In one variable, Ehrhart theory has been vastly generalized. For example it has been established that counting any set that can be defined in Presburger Arithmetic and a single parameter is quasipolynomial! As a corollary, in David's question, the counting function is a quasipolynomial when we specialize the value of one of the variables.

The multiparameter version of Ehrhart theory is bound to get a bit trickier. For example the number of lattice points in the segment between $(s,0)$ and $(0,t)$ is $1+\gcd(s,t)$. This function is a quasipolynomial when you specialize either value, but it is not a quasipolynomial in both variables, meaning that there is no integer $N$ so that when restricting $s,t$ to residue classes $(\text{mod } N)$ we get a polynomial in $s,t$.

My question is motivated from a basic property of polynomials (what Richard Palais calls analogs of Hartog's theorem): If a multivariate function is a polynomial of bounded degree in each variable separately, then it is a multivariable polynomial.

Question: Can we describe bivariate functions that are quasipolynomial of bounded degree in each variable? Are they bivariate quasipolynomials plus perhaps linear combinations of terms like $\gcd(P,Q)$, for bivariate polynomials $P,Q$?

Answer 1

There is a multivariate version of an Ehrhart quasipolynomial (but it is necessarily only piecewise defined). Namely, fix an integer matrix A, the number of integer lattice points in the rational polytope { x : Ax ≤ b } is a piecewise-defined quasipolynomials in the components of b (viewed as an integer vector). Geometrically, what we're doing is fixing the normal vectors of a set of (possible) facets of a polytope and then vary the position of the facets. (Note that in your example, the normals are not preserved.)

It is a short step (using slack variables) to convert ≤ into =, and then the counting function above goes by the name of vector partition function (associated with A). See this seminal paper by Dahmen & Micchelli as a starting point.