On the enumeration of polynomials with prescribed factorization pattern

TL;DR

This paper develops generating function techniques over group rings to precisely count finite field polynomials with specific initial coefficients and factorization patterns, including smoothness and trace constraints.

Contribution

It introduces new exact formulas for counting polynomials with prescribed coefficients and factorization patterns over finite fields, advancing enumeration methods.

Findings

Derived formulas for monic n-smooth polynomials of degree m

Counted polynomials with prescribed trace coefficient

Provided enumeration techniques using generating functions

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.