Counting invariant subspaces and decompositions of additive polynomials

TL;DR

This paper studies the structure and enumeration of decompositions of r-additive polynomials over fields of positive characteristic, linking them to Frobenius-invariant subspaces and providing efficient algorithms for their analysis.

Contribution

It introduces an efficient algorithm to compute the Frobenius automorphism's Jordan form and derives formulas to count decompositions of r-additive polynomials.

Findings

Algorithm for Frobenius automorphism's Jordan form

Formula for counting Frobenius-invariant subspaces

Enumeration of polynomial decompositions

Abstract

The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f(x) = g(h(x)) in F[x] over a field F is well understood in many cases, but less well when the degree of f is divisible by the positive characteristic p of F. This work investigates the decompositions of r-additive polynomials, where every exponent and also the field size is a power of r, which itself is a power of p. The decompositions of an r-additive polynomial f are intimately linked to the Frobenius-invariant subspaces of its root space V in the algebraic closure of F. We present an efficient algorithm to compute the rational Jordan form of the Frobenius automorphism on V. A formula of Fripertinger (2011) then counts the number of Frobenius-invariant subspaces of a given dimension and we derive the number of decompositions with prescribed degrees.