In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and sufficient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This criterion does not require any monomial ordering and generalizes the Buchberger criterion of S-polynomials. It leads to a new algorithm for constructing the multiplicative structure of a zero-dimensional algebra. Described in terms of intrinsic operations on vector spaces in the ring of polynomials, this algorithm extends naturally to Laurent polynomials.

We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde systems and can yield dramatic simplifications of complex overdetermined nonlinear pde systems. Such overdetermined systems arise in analysis of physical pdes for reductions to odes using the Nonclassical Method, the search for exact solutions of Einstein's field equations and the determination of discrete symmetries of differential equations. Application of the algorithm to the associated nonlinear overdetermined system of 856 pdes arising when the Nonclassical Method is applied to a cubic nonlinear Schrodinger system yields new results. Our algorithm combines features of geometric involutiv...

In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and non-multiplicative. Given an admissible ordering, this separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations. Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce, in terms of the latter, the concept of involutivity for polynomial systems. We prove that an involutive system is a special, generally redundant, form of a Gröbner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, those of Janet and Thomas, provide an algorithmic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In particular, we incorporate Buchberger’s chain criterion to avoid unnecessary reductions. The implementation for Pommaret division has been done in Reduce.

A completion procedure for computing a canonical basis for a k-subalgebra is proposed. Using this canonical basis, the membership problem for a k-subalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewriting concepts. A canonical basis produced by the completion procedure shares many properties of a Grobner basis such as reducing an element of a k-subalgebra to 0 and generating unique normal forms for the equivalence classes generated by a k-subalgebra. In contrast to Shannon and Sweedler's approach using tag variables, this approach is direct. One of the limitations of the approach however is that the procedure may not terminate for some term orderings thus giving an infinite canonical basis. The procedure is illustrated using examples.

this paper is to make the similarity between Knuth-Bendix completion and the Buchberger algorithm explicit, by describing a general algorithm called S-normalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations 3 instances of this general algorithm for a particular choice of S. This has been achieved in two steps.

In this paper we describe some experimental findings on selection strategies for Gröbner basis computation with the Buchberger algorithm. In particular, the results suggest that the "sugar flavor" of the "normal selection", implemented first in CoCoA, then in AlPI, and now in SCRATCHPAD-II, is the best choice for a selection strategy. It has to be combined with the "straight-forward" simplification strategy and with a special form of the Gebauer-Möller criteria to obtain the best results. The idea of the "sugar flavor" is the following: the Buchberger algorithm for homogeneous ideals, with degree-compatible term ordering and normal selection strategy, usually works fine. Homogenizing the basis of the ideal is good for the strategy, but bad for the basis to be computed. The sugar flavor computes, for every polynomial in the course of the algorithm, "the degree that it would have if computed with the homogeneous algorithm", and uses this phantom degree (the sugar) only for the selection strategy. We have tested several examples with different selection strategies, and the sugar flavor has proved to be always the best choice or very near to it. The comparison between the different variants of the sugar flavor has been made, but the results are up to now inconclusive. We include a complete deterministic description of the Buchberger algorithm as it was used in our experiments.

DAE and PDAE are systems of ordinary and partial differential-algebraic equations with constraints. They occur frequently in applications such as constrained multibody mechanics, space-craft control and incompressible fluid dynamics. A dae has differential index r if a minimum of r +1 differentiations of it are required before no new constraints are obtained. While dae of low differential index (0 or 1) are generally easier to solve numerically, higher index dae present severe difficulties. Reich, Rabier and Rheinboldt have presented a geometric theory and an algorithm for reducing dae of high differential index to dae of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for dae of low differential index. We show that for analytic autonomous first order dae this algorithm is equivalent to the Cartan-Kuranishi algorithm for completing a system of differential equations to involutive form. The CartanKuranishi algorithm has the advantage tha...

We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's original algorithm for polynomial rings over a field.

This paper is an attempt to address the processing of nonlinear numerical constraints over the Reals by combining three different methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we define a generic two-step constraint processing technique based on an extension of the Constraint Satisfaction Problem, called Extended Constraint Satisfaction Problem (ECSP). The first step is a rewriting step, in which the initial ECSP is symbolically transformed. The second step, called approximation step, is based on a local consistency notion, called weak arc-consistency, defined over ECSPs in terms of fixed point of contractant monotone operators. This notion is shown to generalize previous local consistency concepts defined over finite domains (arc-consistency) or infinite subsets of the Reals (arc B-consistency and interval, hull and box-consistency). A filtering algorithm, derived from AC-3, is given and is shown to be corr...

In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Gröbner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced Gröbner basis. 1