We present an algorithm which converts a given Gröbner basis of a polynomial ideal I to a Gröbner basis of I with respect to another term order. The conversion is done in several steps following a path in the Gröbner fan of I. Each conversion step is based on the computation of a Gröbner basis of a toric degeneration of I. c ○ 1997 Academic Press Limited 1.

Monomial representations and operations for Grobner bases computations are investigated from an implementation point of view. The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Grobner bases. Furthermore, a rank-based monomial representation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorized comparisons. Extensive benchmark tests with the Computer Algebra System Singular are used to evaluate these concepts. 1 Introduction The method of Grobner bases (GB) (see, for example, [8] for an introduction) is undoubtly one of the most important and prominent success stories of the field of Computer Algebra. Starting in the 1960's, an unsolved problem has developed into an essential computational tool with a great variety of applications and more and more powerful implementations. The heart of the GB method are computations of Grobner or Standard bases with...

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.

Abstract. A natural topology on the space of left orderings of an arbitrary semi-group is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. 1. Orderings for semi-groups Given a semi-group G (ie. a set with an associative binary operation), a linear order, <, on G is a left order if a < b implies ca < cb, for any c. Similarly, a linear order,<, is a right order if a < b implies ac < bc, for any c ∈ G. The sets of all left and right orderings of G are denoted by LO(G) and RO(G) respectively. If G is a group then there is a 1-1 correspondence between these two sets which associates with any left ordering, <l, a right ordering, <r, such that a <r b if and only if b −1 <l a −1. For more about ordering of groups see [4, 7, 8]. Let Ua,b ⊂ LO(G) denote the set of all left orderings, <, for which a < b. We can put a topology on LO(G) in one of the following two ways. Definition 1.1. LO(G) has the smallest topology for which all the sets Ua,b are open. Any open set in this topology is a union of sets of the form

Let m be a nonnegative integer, n a positive integer, N = f0; 1; 2; :::g and Nn = f1; : : : ; ng. A ranking is a total order of N m Nn such that (a; i) (b; j) implies (a + c; i) (b + c; j) for a, b, c 2 N m and i; j 2 Nn . We describe an approach to such rankings and a theorem which gives an explicit construction of an arbitrary ranking using nite real data. The case n = 1 corresponds to term-orderings of monomials which are crucial inputs for Buchberger's Grobner Basis algorithm for polynomial rings. The case n > 1 corresponds to rankings of partial derivatives which are inputs in algorithms in dierential algebra and Buchberger's algorithm for free modules over polynomial rings. A subclass of such rankings determined by nite integer data is given which is suÆcient for eective implementation of such rankings. This has been implemented in the symbolic language Maple. The rankings considered by Riquier are a special case of those considered here. Examples including appl...

The problem of choosing efficient algorithms and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Grobner bases is expensive, and in noncommutative algebras is not guaranteed to terminate. The algorithm, together with the order used to determine the leading term of each polynomial, are known to affect the cost of the computation, and are the focus of this thesis. A Gröbner basis is a set of polynomials computed, using Buchberger's algorithm, from another set of polynomials. The noncommutative form of Buchberger's algorithm repeatedly constructs a new polynomial from a triple, which is a pair of polynomials whose leading terms overlap and form a nontrivial common multiple. The algorithm leaves a number of details underspecified, and can be altered to improve its behavior. A significant improvement is the devel...

We study the uniformity of Buchberger algorithms for computing Grobner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problem is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singularity theory. For arbitrary input sets uniformity is in general impossible. By way of contrast we show that the Buchberger algorithm is indeed uniform up to a finite case distinction on the exponential parameter k for inputs consisting of monomials and binomials only. Under this hypothesis the case distinction is algorithmic and partitions the parameter range into Presburger sets. In each case the Buchberger algorithm is uniform and can be described explicitly and algorithmically. In the course of the algorithm the exponential parameter k enters also the coe#cients as exponent. Thus the uniformity in k is established with respect to parametric exponents in both terms and coe#cients. These results are obtained as a consequence of a much more general theorem concerning Buchberger algorithms for sets of monomials and binomials with arbitrary parametric coe#cients and exponents, generalizing the construction of Grobner systems.

In this paper, we present an optimal, exponential space algorithm for generating the reduced Gröbner basis of binomial ideals. We make use of the close relationship between commutative semigroups and pure difference binomial ideals. Based on the algorithm for the uniform word problem in commutative semigroups exhibited by Mayr and Meyer we first derive an exponential space algorithm for constructing the reduced Gröbner basis of a pure difference binomial ideal. In addition to some applications to finitely presented commutative semigroups, this algorithm is then extended to an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals in general.

Recently, Zharkov and Blinkov introduced the notion of involutive bases of polynomial ideals. This involutive approach has its origin in the theory of partial differential equations and is a translation of results of Janet and Pommaret. In this paper we present a pure algebraic foundation of involutive bases of Pommaret type. In fact, they turn out to be generalized left Gröbner bases of ideals in the commutative polynomial ring with respect to a non-commutative grading. The introduced theory will allow not only the verification of the results of Zharkov and Blinkov but it will also provide some new facts.

Abstract. We introduce a preordered version of D. Scott's equilogical spaces [Sc], called inequilogical spaces, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients ' of spaces, which are not topological spaces and are of interest in noncommutative geometry, with finer results than the ones obtained with equilogical spaces, in a previous paper. This setting is compared with other structures which have been recently used for Directed Algebraic Topology: spaces equipped with an order, or a local order, or distinguished paths or distinguished cubes.