There are many good reasons to teach a course on a systematic introduction to symbolic methods not only to students of mathematics but also to those of technical sciences. The design of such a course meets an essential difficulty since the principles to be demonstrated appear only in non trivial applications in a convincing way, but there is no time to teach the necessary contexts to a large extend. Hence the material intended to demonstrate different effects has to be chosen with great care. The goal of this paper is to show that for such a course algebraic numbers are not only interesting by their mathematical content but also as a complex target where different concepts and principles of symbolic computations become apparent. Thus they may serve at once as a non trivial application of the basic concepts, notations and principles developed earlier in such a course.

This paper continues our study of applications of factorized Gröbner basis computations in [8] and [9]. We describe a way to interweave factorized Gröbner bases and the ideas in [5] that leads to a significant speed up in the computation of isolated primes for well splitting examples. Based on that observation we generalize the algorithm presented in [22] to the computation of primary decompositions for modules. It rests on an ideal separation argument. We also discuss the practically important question how to extract a minimal primary decomposition, neither addressed in [5] nor in [17]. For that purpose we outline a method to detect necessary embedded primes in the output collection of our algorithm, similar to [22, cor. 2.22]. The algorithms are partly implemented in version 2.2.1 of our REDUCE package CALI [7].

Let k be a field, S = k[xv : v ϵ V] be the polynomial ring over the finite set of variables (xv : v ϵ V), and m = (xv : v ϵ V) the ideal defining the origin of Spec S. It is theoretically known (see e.g. Alonso et el., 1991) that the algorithmic ideas for the computation of ideal (and module) intersections, quotients, deciding radical membership etc. in S may be adopted not only for computations in the local ring Sm but also for term orders of mixed type with standard bases replacing Gröbner bases. Using the generalization of Mora's tangent cone algorithm to arbitrary term orders we give a detailed description of the necessary modifications and restrictions. In a second part we discuss a generalization of the deformation argument for standard bases and independent sets to term orders of mixed type. For local term orders these questions were investigated in Gräbe (1991). The main algorithmic ideas described are implemented in the author's REDUCE package CALI (Gräbe, 1993a).

Systems of polynomial equations often have symmetries. In solving such a system using Buchberger's algorithm, the symmetries are neglected. Incorporating symmetries into the solution process enables us to solve larger problems than with Buchberger's algorithm alone. This paper presents a method that shows how this can be achieved and also gives an algorithm that brings together continuously parameterized symmetries with Buchberger's algorithm.

Operators which map subsets of a given set to the set of their minimal elements with respect to some relation R form the basis of a semantic approach in non-monotonic logic, belief revision, conditional logic and updating. In this paper we investigate operators of this type from an algebraic viewpoint. A representation theorem is proved and various properties of the resulting algebras are investigated. It is shown that they behave quite differently from known algebras related to logics, e.g. modal algebras and Heyting algebras.

We report on some experience with a new version of the well known Gröbner algorithm with factorization and constraint inequalities, implemented in our REDUCE package CALI, [12]. We discuss some of its details and present run time comparisons with other existing implementations on well splitting examples.

The goal of this paper is to transfer convolution, correlation and Fourier transform to second order tensor fields. Convolution of two tensor fields is defined using matrix multiplication. Convolution of a tensor field with a scalar mask can thus be described by multiplying the scalars with the real unit matrix. The Fourier transform of tensor fields defined in this paper corresponds to Fourier transform of each of the tensor components in the field. It is shown that for this convolution and Fourier transform, the well known convolution theorem holds and optimization in speed can be achieved by using Fast Fourier transform algorithms. Furthermore, pattern matching on tensor fields based on this convolution is described.

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.

In a preceding paper [9] we reported on some experience with a new version of the well known Gröbner algorithm with factorization and constraint inequalities. Here we discuss, how this approach may be refined to produce triangular systems in the sense of [12] and [13]. Such a refinement guarantees, different to the usual Gröbner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals. As in [9] our method weakens the usual restriction to lexicographic term orders. Triangular systems are a very helpful tool between factorization at a heuristical level and full decomposition into prime components. Our approach grew up from a consequent interpretation of the algorithmic ideas in [5] as a delayed quotient computation in favour of early use of (multivariate) factorization. It is implemented in version 2.2 of the REDUCE package CALI [8].

We introduce a special set of relations on a finite set, called clausal relations. A restricted version of the Galois connection between polymorphisms and invariants, called Pol-CInv, is studied, where the invariant relations are clausal relations. Clones arising from this Galois connection, so-called C-clones, are investigated. Finally, we show that clausal relations meet a sufficient condition that is known to ensure polynomial time solvability of the corresponding CSP.