Computing zeta functions of large polynomial systems over finite fields

TL;DR

This paper presents improved algorithms for computing zeta functions of large polynomial systems over finite fields, reducing complexity from exponential to polynomial dependence on the number of equations, enabling faster computations.

Contribution

It introduces a polynomial-time dependence on the number of equations for zeta function computation, improving upon previous exponential algorithms.

Findings

Reduced the dependence on the number of equations from exponential to polynomial.

Speeded up a doubly exponential time algorithm to singly exponential time.

Developed an effective version of the Kronecker theorem for large polynomial systems.

Abstract

In this paper, we improve the algorithms of Lauder-Wan \cite{LW} and Harvey \cite{Ha} to compute the zeta function of a system of $m$ polynomial equations in $n$ variables over the finite field $\FF_q$ of $q$ elements, for $m$ large. The dependence on $m$ in the original algorithms was exponential in $m$. Our main result is a reduction of the exponential dependence on $m$ to a polynomial dependence on $m$. As an application, we speed up a doubly exponential time algorithm from a software verification paper \cite{BJK} (on universal equivalence of programs over finite fields) to singly exponential time. One key new ingredient is an effective version of the classical Kronecker theorem which (set-theoretically) reduces the number of defining equations for a "large" polynomial system over $\FF_q$ when $q$ is suitably large.