Joachim Apel

An extension of Buchberger's algorithm and calculationsin enveloping fields of lie algebras

The powerful concept of Grobner bases and an extension of the Buchberger algorithm for their computation have been generalised to enveloping algebras of Lie algebras. Algorithms for the computation of syzygies by use of Grobner bases are given. This is the first method which allows the transformation of right fractions into left fractions in the Lie field of any finite dimensional Lie algebra. That enables CAS calculations in Lie fields. An AMP program LIEFIELD has been written for this purpose. Another AMP program SYZYGY produces a generating set of the syzygy module of any finite subset of an enveloping algebra. Examples for both programs are presented for the Weyl algebra and the so(3).

The Theory of Involutive Divisions and an Application to Hilbert Function Computations

Generalising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more flexible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov.Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Grobner basis and that it is straightforward to compute the (affine) Hilbert function of an idealIfrom an arbitrary involutive basis of alI.

On a Conjecture of R. P. Stanley; Part II--Quotients Modulo Monomial Ideals

In 1982 Richard P. Stanley conjectured that any finitely generated ℝn-graded module M over a finitely generated ℕn-graded K-algebra R can be decomposed as a direct sum M = ⊕i = 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M.We will study this conjecture for modules M = R/I, where R is a polynomial ring and I a monomial ideal. In particular, we will prove that Stanleys Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.

Detecting unnecessary reductions in an involutive basis computation

We consider the check of the involutive basis property in a polynomial context. In order to show that a finite generating set F of a polynomial ideal I is an involutive basis one must confirm two properties. Firstly, the set of leading terms of the elements of F has to be complete. Secondly, one has to prove that F is a Grobner basis of I. The latter is the time critical part but can be accelerated by application of Buchbergers criteria including the many improvements found during the last two decades. Gerdt and Blinkov [Gerdt, V.P., Blinkov, Y.A., 1998. Involutive bases of polynomial ideals. Mathematics and Computers in Simulation 45, 519-541] were the first who applied these criteria in involutive basis computations. We present criteria which are also transferred from the theory of Grobner bases to involutive basis computations. We illustrate that our results exploit the Grobner basis theory slightly more than those of Gerdt and Blinkov. Our criteria apply in all cases where those of Gerdt/Blinkov do, but we also present examples where our criteria are superior. Some of our criteria can also be used in algebras of solvable type, e.g., Weyl algebras or enveloping algebras of Lie algebras, in full analogy to the Grobner basis case. We show that the application of criteria enforces the termination of the involutive basis algorithm independent of the prolongation selection strategy.

Computational ideal theory in finitely generated extension rings

Since Buchberger introduced the theory of Grobner bases in 1965 it has become an important tool in constructive algebra and, nowadays, Buchbergers method is fundamental for many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings a lot of possibilities to generalize the ideas to other types of rings have been investigated. The perhaps most general concept, though it does not cover all possible extensions, is the theory of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Grobner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Grobner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Grobner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Grobner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results.

On a Conjecture of R. P. Stanley; Part I—Monomial Ideals

In 1982 Richard P. Stanley conjectured that any finitely generated ℝn-graded module M over a finitely generated ℕn-graded K-algebra R can be decomposed in a direct sum M = ⊕i = 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M.We will study this conjecture for the special case that R is a polynomial ring and M an ideal of R, where we encounter a strong connection to generalized involutive bases. We will derive a criterion which allows us to extract an upper bound on depth M from particular involutive bases. As a corollary we obtain that any monomial ideal M which possesses an involutive basis of this type satisfies Stanleys Conjecture and in this case the involutive decomposition defined by the basis is also a Stanley decomposition of M. Moreover, we will show that the criterion applies, for instance, to any monomial ideal of depth at most 2, to any monomial ideal in at most 3 variables, and to any monomial ideal which is generic with respect to one variable. The theory of involutive bases provides us with the algorithmic part for the computation of Stanley decompositions in these situations.

A Gröbner Approach to Involutive Bases

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 Grobner 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.

The Computation of Gröbner Bases Using an Alternative Algorithm

When Zharkov and Blinkov ([ZB93]) applied the classical ideas of involutive systems originating from the theory of partial differential equations (c.f. [Ja29], [Po78]) to the computation of Grobner bases (c.f. [Bu65], [BW93]) their theory seemed to be a rather marginal concept. But due to the opportunity of gaining a faster version for one of the most frequently applied algorithms the method came into the focus of computer algebra research (c.f. [Ap95], [GB95], [GS95], [Ma95]). It turned out that Pommaret bases are not only of interest for fast implementations (c.f. [ZB93]) but that they are also a point of contact of different theories which were investigated intensively for a long time. So, the theory of Pommaret bases enables the exchange of useful ideas between the theories as well as it benefits itself from the relationships. A certain similarity of the Zharkov/Blinkov method and the Kandri-Rody/Weispfenning closure technique motivates the study of commutative polynomial rings from a non-commutative point of view. The theory of Pommaret bases can be presented in an algebraic way using the Grobner theory of graded structures. Here we will present the straight forward generalization of Pommaret bases to the class of algebras of solvable type. Under the non-commutative grading most calculations are pushed back to the free non-commutative polynomial ring. This provides a link to the theory of term rewriting and the Zharkov/Blinkov method appears as an application of the prefix reduction/saturation technique of Madlener and Reinert (c.f.[MR93]) with a restricted saturation. The restricted saturation has its natural origin in the syzygy theory and heavily improves the termination behaviour in the particular case of Pommaret bases. So, it seems to be worth to investigate the effect of splitting the saturation step also for similar term rewriting problems.

Reduction of everywhere convergent power series with respect to Gröbner bases

Abstract We introduce a notion of Grobner reduction of everywhere convergent power series over the real or complex numbers with respect to ideals generated by polynomials and an admissible term ordering. The presented theory is situated somewhere between the known theories for polynomials and formal power series. Our main theorem states the existence of a formula for the division of everywhere convergent power series over the real or complex numbers by a finite set of polynomials. If the set of polynomials is a Grobner basis then the remainder of that division depends only on the equivalence class of the power series modulo the ideal generated by the polynomials. When the power series which shall be divided is a polynomial the division formula leads to a usual Grobner representation well known from polynomial rings. Finally, the results are applied to prove the closedness of ideals generated by polynomials in the ring of everywhere convergent power series and to give a very simple proof of the affine version of Serres graph theorem.

FELIX—an assistant for alebraists

FELIX is a special computer algebra system designed for calculations with elements of algebraic structures as well as with substructures and homomorphisms. It covers both commutative polynomial rings and modules and non-commutative structures. Buchberger’s algorithm for the computation of Grobner bases is fundamental for many of the included operations. The article contains a short description of the system FELIX and illustrates the sensitivity y of Buchberger’s algorithm against changes of selection strategies.