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

## TL;DR

This paper proves that for algebraic sets over finite fields, the number of defining equations can be reduced to the dimension of the space without increasing degrees, simplifying the description of these sets.

## Contribution

It introduces a method to reduce the number of defining equations for algebraic sets over finite fields, maintaining degrees and providing a new way to simplify algebraic descriptions.

## Key findings

- Number of equations can be reduced to the space dimension
- Reduction preserves the total degree of defining polynomials
- Applicable to both affine and projective algebraic sets

## 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.