Parameterized Type Definitions in Mathematica: Methods and Advantages

TL;DR

This paper introduces a parameterized categorical approach to symbolic algebra in Mathematica, enabling flexible, extendable polynomial computations and Groebner basis algorithms through a new type system.

Contribution

It develops a multivariate polynomial category with parameterization principles, enhancing Mathematica's capabilities with a novel, extendable type system inspired by prior work in Maple.

Findings

Implemented Groebner basis algorithms in Mathematica using the new type system.

Demonstrated advantages like operation protection, inheritance, and extendibility.

Extended Mathematica with a practical, flexible polynomial type system.

Abstract

The theme of symbolic computation in algebraic categories has become of utmost importance in the last decade since it enables the automatic modeling of modern algebra theories. On this theoretical background, the present paper reveals the utility of the parameterized categorical approach by deriving a multivariate polynomial category (over various coefficient domains), which is used by our Mathematica implementation of Buchberger's algorithms for determining the Groebner basis. These implementations are designed according to domain and category parameterization principles underlining their advantages: operation protection, inheritance, generality, easy extendibility. In particular, such an extension of Mathematica, a widely used symbolic computation system, with a new type system has a certain practical importance. The approach we propose for Mathematica is inspired from D. Gruntz and M. Monagan's work in Gauss, for Maple.