On local linear convexity generalized to commutative algebras

TL;DR

This paper generalizes the concept of local linear convexity from complex spaces to Cartesian products of commutative associative algebras with invertible basis elements, providing new conditions for convexity in this algebraic setting.

Contribution

It introduces a framework for analyzing local A-linear convexity in spaces formed by Cartesian products of commutative algebras, extending classical convexity notions.

Findings

Necessary and sufficient conditions for local A-linear convexity are established.

Conditions are expressed via nonnegativity and positivity of formal quadratic differential forms in A.

The work generalizes properties of linearly convex domains to algebraic structures.

Abstract

A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there is at least one that is non-degenerate. The notion of linearly convex domains in the multi-dimensional complex space and some of their properties are generalized to the space that is the Cartesian product of n algebras A. Namely, the separate necessary and sufficient conditions of the local A-linear convexity of domains with smooth boundary in the space are obtained in terms of nonnegativity and positivity of formal quadratic differential form in A, respectively.