Fundamental algebraic sets and locally unit-additive rings

TL;DR

This paper generalizes the concept of the Fundamental Theorem of Algebra to a broader algebraic geometric setting, introducing fundamental algebraic sets and locally unit-additive rings, and establishing their equivalence.

Contribution

It introduces the notions of fundamental algebraic sets and local fundamentality, extending the concept of unit-additivity to local contexts and linking these properties to affine varieties.

Findings

Affine variety is fundamental iff its coordinate ring is unit-additive.

Many equivalent definitions of local unit-additivity are established.

Examples illustrate the concepts throughout the paper.

Abstract

The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.