Gr\"obner Bases with Reduction Machines

TL;DR

This paper introduces reduction machines for computing Gr"obner bases, allowing arbitrary monomial choices during polynomial reduction, and demonstrates their correctness and efficiency through implementation and experiments.

Contribution

It proposes reduction machines that perform polynomial reduction with arbitrary monomial choices, ensuring consistent normal forms and improving computational efficiency.

Findings

Reduction machines can simulate any polynomial reduction process.

Normal forms are independent of the reduction order with fixed reductors.

Experimental results show the effectiveness of the proposed implementations.

Abstract

In this paper, we make a contribution to the computation of Gr\"obner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we investigate what happens if we make that choice arbitrarily. It turns out not only this is possible (the fact that this produces a normal form being already known in the literature), but, for a fixed choice of reductors, the obtained normal form is the same no matter the order in which we reduce the monomials. To prove this, we introduce reduction machines, which work by reducing each monomial independently and then collecting the result. We show that such a machine can simulate any such reduction. We then discuss different implementations of these machines. Some of these implementations address inherent inefficiencies in reduction machines (repeating the same computations). We describe a first implementation and look at some experimental results.