The distribution of defective multivariate polynomial systems over a finite field

2022/08/17 by Nardo Giménez, Guillermo Matera, Giménez, Nardo +5
Computer Science · Mathematics · #11G25 #14G05 #14G15 #14M10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2208.08572

openalex publication_date 2022/08/17 · openalex created_date 2022/08/20 · openalex updated_date 2026/07/28

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.

Related

• Explicit estimates for polynomial systems defining irreducible smooth complete intersections
• The polynomial method over varieties
• Counting decomposable multivariate polynomials
• Number of irreducible polynomials in several variables over finite fields
• The number of solutions of a random system of polynomials over a finite field