Bases of subalgebras of K[[x]] and K[x]

TL;DR

This paper investigates the structure of subalgebras generated by formal power series or polynomials, focusing on their semigroups of orders or degrees, and provides algorithms for their computation with various applications.

Contribution

It introduces procedures to compute semigroups of orders and degrees for subalgebras of formal power series and polynomials, with new applications.

Findings

Algorithms for computing semigroups of orders and degrees

Characterization of subalgebra structures

Applications to algebraic and computational problems

Abstract

Let $f\_1,\ldots, f\_s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[ f1,\ldots, f\_s]] \subseteq K[[ x ]]$ (respectively the semigroup of degrees of polynomials in$K[f\_1,\ldots,f\_s]\subseteq K[x]$). We give procedures to compute thesesemigroups and several applications.