1 Introduction

The theorem that any algebraic set in \(n\)-dimensional space is the intersection of \(n\) hypersurfaces ¹ has been proved independently by Storch ( [ 1 ] ), and Eisenbud and Evans ( [ 2 ] ); both short proofs are ring-theoretic, i.e., one reduces the number of generators of radical ideals.

In this note we examine closer the finite fields case of the problem. If just the number of equations needed to describe an algebraic set is in question, then the answer is immediate: it is easy to construct a single polynomial defining it. If, however, the nature of defining polynomials (e.g., their total degree) is to be preserved, this problem becomes more interesting.

It turns out that we can avoid dealing with rings; the vector space structure is sufficient and, as in the theorem cited above, our result again produces \(n\) equations; moreover, we show that these new equations can be chosen to be linear combinations with scalar coefficients of the old ones, so, roughly speaking, they remain of the same type (see Corollaries 3 and 4, with accompanying examples), and our proof is surprisingly elementary.