Membership problem, Representation problem and the Computation of the Radical for one-dimensional Ideals

1997

Journal of Complexity

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g 1 , g 2 , . . . , g w ) where f and the g i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.

“…For earlier, somewhat weaker doubly exponential degree bounds, also see [25] and [37]. Similar results, but for more restricted subproblems, can also be found in e.g.…”

Section: Basic Resultsmentioning

confidence: 56%

Some Complexity Results for Polynomial Ideals

1997

Journal of Complexity

“…[16]) which are double exponential. Such time bounds can be derived from the space bounds by the reasoning from section 3.3.…”

Section: Resultsmentioning

confidence: 98%

Space-efficient Gröbner basis computation without degree bounds

Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

Ritscher

2011

The computation of a Gröbner basis of a polynomial ideal is known to be exponential space complete. We revisit the algorithm by Kühnle and Mayr using recent improvements of various degree bounds. The result is an algorithm which is exponential in the ideal dimension (rather than the number of indeterminates).Furthermore, we provide an incremental version of the algorithm which is independent of the knowledge of degree bounds. Using a space-efficient implementation of Buchberger's S-criterion, the algorithm can be implemented such that the space requirement only depends on the size of the representation and the Gröbner basis degrees of the problem instance (instead of the worst case), and thus is much lower in average.

“…Extending this result, Krick and Logar (1991) showed that Gröbner bases of zero-or one-dimensional ideals can be computed in time exponential in the number of indeterminates.…”

Section: Introductionmentioning

confidence: 94%

An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals, and Applications to Commutative Semigroups

Koppenhagen

Journal of Symbolic Computation

2001

It is known that the reduced Gröbner basis of general polynomial ideals can be computed in exponential space. The algorithm, obtained by Kühnle and Mayr, is, however, based on rather complex parallel computations, and, above that, makes extensive use of the parallel computation thesis. In this paper, we exhibit an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals which can be implemented without any complex parallel computations. This result is then applied to derive space optimal decision procedures for the finite enumeration and subword problems for commutative semigroups.