Grobner bases and extension of scalars

TL;DR

This paper investigates how Grobner bases behave under scalar extensions, establishing conditions for their compatibility and exploring how they can inform about fibers and aid in algorithm development.

Contribution

It proves that scalar extension commutes with Grobner bases if and only if the extension is flat, and explores applications in algebraic geometry and algorithm design.

Findings

Scalar extension commutes with Grobner bases iff the extension is flat

Grobner bases can reveal fiber information of algebraic families

Algorithms can be developed to find loci of finiteness or isomorphisms

Abstract

This paper studies the behavior of Grobner bases with respect to extensions of scalars. We prove that an extension of scalars commutes with taking Grobner bases iff the extension is flat. We consider what information can be deduced about fibers of a family, from the Grobner basis of the defining ideal of the family itself. This information can be used to construct algorithms, such as to find locii over which a map is finite, or an isomorphism.