Counting decomposable multivariate polynomials

TL;DR

This paper investigates the structure and enumeration of decomposable multivariate polynomials, providing dimension results over algebraically closed fields and accurate approximations over finite fields with exponentially decreasing error.

Contribution

It determines the dimension of the set of decomposable multivariate polynomials and offers precise approximations for their count over finite fields.

Findings

Dimension of decomposable polynomials over algebraically closed fields is established.

Accurate approximation of the number of such polynomials over finite fields with exponentially decreasing error.

Provides theoretical insights into the structure of decomposable multivariate polynomials.

Abstract

A polynomial f (multivariate over a field) is decomposable if f = g(h) with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The relative error in our approximations is exponentially decaying in the input size.