On the enumeration of polynomials with prescribed factorization pattern

2021/05/17 by Kuttner, Simon, Wang, Qiang
#11T06 #12E05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.07550

Abstract

We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic n-smooth polynomial of degree m over a finite field, as well as the number of monic n-smooth polynomial of degree m with the prescribed trace coefficient.

Related

• Counting irreducible polynomials with prescribed coefficients over a finite field
• The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace
• Counting polynomials over finite fields with given root multiplicities
• Factorization patterns on nonlinear families of univariate polynomials over a finite field
• The distribution of factorization patterns on linear families of polynomials over a finite field