Gr\"obner Bases under Composition, Survey paper

TL;DR

This survey reviews research from 1995 to 2007 on how polynomial composition affects the computation of Gr"obner bases across various settings, highlighting known results and open questions.

Contribution

It consolidates existing knowledge on Gr"obner bases under composition, identifying completed and open problems, and discusses generalizations and special cases.

Findings

Collected key results on Gr"obner bases under composition

Identified open questions and areas for further research

Summarized advances over different polynomial rings and orderings

Abstract

This paper is a survey on the study of the behaviour of the composition of polynomials on the computation of Gr\"obner bases. This survey brings together some works published between 1995 and 2007. The authors of these papers gave answers to some questions in this subject for several types of Gr\"obner bases, over different monomials orderings and over different polynomials rings. Some of these answers are complete and some are not. Some papers of them were given to answer some open questions given in the first published paper of these papers and other some were given to generalize previous work. Note that, there are many other works in this subject over other special cases and types of Gr\"obner bases under the usual definition over arbitrary fields, over other types of bases, under the non-commutative case and with other methods of proofs and calculations.