Generalizations of two theorems of Ritt on decompositions of polynomial maps

TL;DR

This paper extends Ritt's theorems on polynomial decompositions to broader algebraic structures called reduction monoids, including odd polynomials, and explores their limitations in more complex monoids like the cusp.

Contribution

It generalizes Ritt's theorems to reduction monoids and identifies conditions where these theorems hold or fail, especially for odd polynomials and the cusp.

Findings

Ritt's theorems are valid for certain reduction monoids like (K[x], ◦) and (K[x^2]x, ◦)

Analogues of Ritt's theorems hold for odd polynomials in (K[x^2]x, ◦)

Ritt's theorems fail for the cusp (K+K[x]x^2, ◦) except for decompositions of maximal length

Abstract

Two theorems of J. F. Ritt on decompositions of polynomials maps are generalized to a more general situation: for, so-called, reduction monoids ($(K[x], \circ)$ and $(K[x^2]x, \circ)$ are examples of reduction monoids). In particular, analogues of the two theorems of J. F. Ritt hold for the monoid $(K[x^2]x, \circ)$ of odd polynomials. It is shown that, in general, the two theorems of J. F. Ritt fail for the cusp $(K+K[x]x^2, \circ)$ but their analogues are still true for decompositions of maximal length of regular elements of the cusp.