Complexity of counting integer roots of multivariate polynomials in a polyhedron? Deciding integer roots of multivariate polyomials is undecidable. However what is known about counting integer roots of multivariate polynomials  in $\mathbb Z[x_1,\dots,x_m]$ with both $m$ and total degree fixed and we only look at integer roots in a bounded polyhedron?
The decision version of this problem is obviously in $\mathsf{NP}$, and Manders &amp; Adleman showed that a specific case is NP-complete. Namely, even deciding whether there exists an integer $x \in [0, \gamma]$ such that $x^2 \cong \alpha \mod \beta$ (the input here is the triple $(\alpha,\beta,\gamma)$) is NP-complete, which is only one variable and degree 2. I haven't checked the details in a while, but I'm pretty sure their reduction is parsimonious; if so, the counting version you ask about will be $\mathsf{\# P}$-complete as well.