Gröbner Bases with Reduction Machines

Georgiana Şurlea
(Department of Computer Science, West University. Timişoara, Romania)
Adrian Crăciun
(Department of Computer Science, West University. Timişoara, Romania)

In this paper, we make a contribution to the computation of Gröbner 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.

In Mircea Marin and Adrian Crăciun: Proceedings Third Symposium on Working Formal Methods (FROM 2019), Timişoara, Romania, 3-5 September 2019, Electronic Proceedings in Theoretical Computer Science 303, pp. 61–75.
Published: 2nd September 2019.

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