Cyclotomic enumeration of polynomials

TL;DR

This paper employs cyclotomic identities to count and analyze tuples of polynomials over finite fields, providing explicit formulas for gcd conditions and multiplicity constraints.

Contribution

It introduces new explicit formulas for counting polynomial tuples with specific gcd and multiplicity properties using cyclotomic identities.

Findings

Derived formulas for counting gcd rth power free polynomial tuples

Computed the number of polynomials with multiplicities in generated monoids

Extended enumeration techniques to polynomials over finite fields

Abstract

Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of polynomials such that their greatest common divisor is rth power free. We also compute the number of monic polynomials where the multiplicity of each irreducible factor belongs to the monoid generated by two relatively prime integers.