Computing Gröbner Bases for Path Algebras in C++

Abstract

Given some path algebra P and an ideal of this path algebra I, generated by some set of polynomials f_1,...,f_l, it is very common to want to know, given some new g, if g is in I. Normally, to do this you would have to find some combination of elements of I that gives g, but you could also compute a Gröbner basis G of I, which is characterized as a collection of g_1,...,g_k in I such that I = <G> and <LT(I)> = <LT(G)>, where LT(S) is the set of lead terms of a set of polynomials S. Computing these using Buchberger's Algorithm can be time consuming by hand and occasionally is actually non-terminating, so it would be convenient if we had an immediate way easily generate ideals and compute their corresponding Gröbner bases. This paper will follow our approach to this problem using C++ and serves as documentation to the project found here (https://github.com/xZecora/path-algebras) on Github.