Reducing the number of equations defining a subset of the $n$-space over a finite field

Abstract

Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with scalar coefficients of $f_{1}, \ldots, f_{k}$, defining the same algebraic set. In particular, one reduces the number of equations without increasing the total degree. We also have the corresponding result for systems of homogeneous polynomials defining algebraic sets in projective spaces.