A dynamic algorithm for Gröbner basis computation

Reducing the size and number of linear programs in a dynamic Gröbner basis algorithm

“…We typically denote polynomials by f , g, h, p, q, and r, and the ideal of R generated by any F ⊆ R as F . Following [9], we call a product of powers of the variables of R a term, and a product of a term and an element of K a monomial. We typically denote constants by letters at the beginning of the alphabet, and terms by t, u, v. We denote the exponent vector of a term by its name in boldface; so, if n = 4 and t = x An ordering σ on the set T n of all terms of R is admissible if it is a wellordering that is compatible with divisibility and multiplication; by "compatible with divisibility," we mean that t | u and t = u implies that t < σ u, and by "compatible with multiplication," we mean that t < σ u implies that tv < σ uv.…”

“…Indeed, this question was posed nearly twenty years ago, and was studied both theoretically and practically in two separate papers [9] [21]. The first considered primarily questions of discrete geometry; the dynamic algorithm spills out as a nice application of the ideas, but the authors did not seriously consider an implementation.…”

“…The temporary solution of [9] was to switch the refiner off at a predetermined point. Aside from the obvious drawback that this prevents useful refinement after this point, it also forces unnecessary refinement before it!…”

“…We have tried to follow the vocabulary and notation of [9], with some modifications. Section 2.1 reviews the traditional theory of Gröbner bases, inasmuch as it pertains to the (static) Buchberger algorithm.…”

A Gröbner Basis Approach to CNF-Formulae Preprocessing

“…Computing a Gröbner basis using a Buchberger-variant algorithm can exhibit exponential worst-case time and computational complexity [19], and for this reason that many different techniques have been developed [15] [18] [20] [21] [22] for improving the algorithm, often for specific applications. However, while these techniques deal with the inner-workings of algorithm itself, outside the algorithm there is less control.…”