A parallelization of the Buchberger algorithm

Author(s):  
R. Bradford
Keyword(s):  
2014 ◽  
Vol 24 (08) ◽  
pp. 1157-1182 ◽  
Author(s):  
Roberto La Scala

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.


Acta Numerica ◽  
1997 ◽  
Vol 6 ◽  
pp. 399-436 ◽  
Author(s):  
T. Y. Li

Let P(x) = 0 be a system of n polynomial equations in n unknowns. Denoting P = (p1,…, pn), we want to find all isolated solutions offor x = (x1,…,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Gröbner bases, are the classical approach to solving (1.1), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.


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