The distribution of defective multivariate polynomial systems over a finite field

TL;DR

This paper investigates the properties and bounds of algebraic varieties defined by deficient multivariate polynomial systems over finite fields, focusing on dimensions and counts of such systems.

Contribution

It provides improved bounds on the dimension and number of deficient polynomial systems for specific types of algebraic varieties over arbitrary fields.

Findings

Enhanced bounds on the dimension of deficient systems.

Improved upper bounds on the number of such systems over finite fields.

Results applicable to ideal-theoretic complete intersections and absolutely irreducible varieties.

Abstract

This paper deals with properties of the algebraic variety defined as the set of zeros of a "deficient" sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely irreducible varieties. For these types, we establish improved bounds on the dimension of the set of deficient systems of each type over an arbitrary field. On the other hand, we establish improved upper bounds on the number of systems of each type over a finite field.