Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields

TL;DR

This paper develops counting methods for special classes of multivariate polynomials over finite fields, providing exact formulas and approximations with exponentially decreasing errors.

Contribution

It introduces novel counting techniques for reducible, s-powerful, and relatively irreducible multivariate polynomials over finite fields using generating functions and combinatorial methods.

Findings

Exact formulas for polynomial counts

Approximate counts with exponentially decreasing errors

Enhanced understanding of polynomial structures over finite fields

Abstract

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.