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Some Complexity Results for Polynomial Ideals
, 1997
"... 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, g1, g2,.. ..."
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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, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. 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.
A Superexponential Lower Bound for Gröbner Bases and ChurchRosser Commutative Thue Systems
, 1986
"... The complexity of the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis or a given commutative Thue system into a ChurchRosser system is presently unknown. In this paper we derive a doubleexponential lower bound (22") for the production length and cardinali ..."
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The complexity of the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis or a given commutative Thue system into a ChurchRosser system is presently unknown. In this paper we derive a doubleexponential lower bound (22") for the production length and cardinality of ChurchRosser commutative Thue systems, and the degree and cardinality of Gröbner bases.
New decision algorithms for finitely presented commutative semigroups
 COMPUT. MATH. APPL
, 1981
"... A new solution of the uniform word problem for finitely presented commutative semigroups is constructed from a completion procedure for commutativeassociative term rewriting systems. The corn: pletion procedure transforms a finite presentation into a uniformly terminating equivalence class term re ..."
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A new solution of the uniform word problem for finitely presented commutative semigroups is constructed from a completion procedure for commutativeassociative term rewriting systems. The corn: pletion procedure transforms a finite presentation into a uniformly terminating equivalence class term rewriting system which is ChurchRosser (terminates uniquely) and therefore decides equivalence of words in the given finitely presented commutative semigroup. Words are expressed in multiplicative exponential form, i.e. as finite vectors, so that fixed uniformly terminating Church.Rosser equivalence class term rewriting systems decide equivalence of words in constant space. Since the uniform word problem for finitely presented commutative semigroups requires exponential space on infinitely many instances, a ChurchRosser term rewriting system must be exponentially larger than its presentation for infinitely many presentations. This solution of the uniform word problem for finitely presented commutative semigroups, in addition to being conceptually simpler than previous solutions, is another small step towards the systematic application of uniformly terminating ChurchRosser term rewriting systems to the solved and open decision problems of algebra.
On Polynomial Ideals, Their Complexity, and Applications
, 1995
"... 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 over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known th ..."
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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 over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known that this problem is exponential space complete.