On canonical bases of a formal ${\mathbb K}-algebra

TL;DR

This paper investigates canonical bases of certain subalgebras of formal power series rings over a field, introducing a canonical fan that generalizes the standard fan of an ideal, providing a new geometric perspective.

Contribution

It introduces the concept of a canonical fan for subalgebras of formal power series rings, extending the classical notion of standard fans of ideals.

Findings

Defined the canonical fan for subalgebras of formal power series rings.

Established properties and structure of the canonical fan.

Generalized the concept of standard fans to a broader algebraic setting.

Abstract

We study canonical bases of a subalgebra ${\bf A}={\mathbb K}[\![f_1,\dots,f_s]\!]\subseteq {\mathbb K}[\![x_1,\dots,x_n]\!]$ over a field ${\mathbb K}$, and we associate with ${\bf A}$ a fan called the canonical fan of $\bf A$. This generalizes the notion of the standard fan of an ideal.