Polynomial and Hyperplane Avoidance over the Integers with Applications to Integer-Point Counting
Grigorii Dakhno, Dmitry Gribanov, Nikita Kasianov, Anastasiia Kats, Andrey Kupavskii, Nikita Kuz'min, Stanislav Moiseev
We study the \emph{Polynomial Avoidance Problem}: given a finite collection $F \subseteq Z[x_1,\dots,x_n]$ of nonzero polynomials, find a vector $x \in Z^n$ of minimum $\ell_p$-norm such that $f(x)\neq0$ for every $f \in F$. Each polynomial is given sparsely as a list of its nonzero terms. The affine special case is the \emph{Hyperplane Avoidance Problem}. This special case arises in Barvinok-type algorithms for counting integer points in polyhedra, when certain rational generating functions must be evaluated at a removable singularity. Our main results are as follows: 1) Let $D:=\sum_{f\in F} \text{deg} f$. Every instance admits a feasible vector $x$ satisfying \[ \|x\|_1 \leq \min\left\{D,\frac{D+n}{2}\right\}. \] This bound is tight. Such a vector can be found by a deterministic algorithm whose bit complexity is polynomial in $D$ and the sparse encoding length of the input. Our proof uses a new algorithmic variant of the \emph{Combinatorial Nullstellensatz}; 2) In the affine case, with $m$ input polynomials, such a vector can be constructed by a fast deterministic polynomial-time algorithm using $O(nm)$ arithmetic operations. For the homogeneous case, Barvinok's original approach provides only the weaker guarantee $\|x\|_1 = O((nm)^{n-1})$; 3) The problem is NP-hard for every $p\in R_{\geq 1}\cup\{\infty\}$, even for homogeneous linear forms; 4) As an application, we improve the complexity bound for counting integer points in general rational polytopes. We also improve the bound for polyhedra of bounded codimension; this includes the counting version of the \emph{Unbounded Subset-Sum} problem as a special case.
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