Finding roots of a multivariate polynomial in a linear subspace

Abstract

Suppose F is a polynomial of total degree d in t variables over a finite field $k={\mathbb{F}}_{q^n}$. We are interested in finding roots of F that lie in a ${\mathbb{F}}_q$-linear subspace of $k^t$. For $m\le n$, we characterize a large class of m-dimensional ${\mathbb{F}}_q$-subspaces U of $k^t$ such that the set of roots of F that lie in U can be bounded by $d^m$ in cardinality, independent of q, and constructed in expected time polynomial in n, t and $d^m$.