Franz Pauer

On lucky ideals for Gro¨bner basis computations

Abstract Let R be a principal ideal ring (i.e. a commutative ring such that all ideals are principal; we do not assume that R is entire), R[x]: = R[x1, …, xn] the polynomial ring in n variables over R and I an ideal in R[x]. Intuitively, an ideal P …; R is “lucky” for I, if we do not loose too much information on Grobner bases of I, when we project I to R [x]: = ( R P [x] . Let F be an ideal basis of I. Examples in Ebert (1983) show that the coefficients of F give no direct criterion to detect luckiness of P, even if R is a domain, K its quotient field and F a Grobner basis of the ideal generated by F in K[x]. However, If we consider the Grobner basis of the ideal generated by F in R[x], we get direct and full information about lucky ideals. The main objective of this article is to give a precise version of this observation. As an application, a short proof for the main result in Winkler (1988) is given.

Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations

Abstract We develop a basic theory of Gröbner bases for ideals in the algebra of Laurent polynomials (and, more generally, in its monomial subalgebras). For this we have to generalize the notion of term order. The theory is applied to systems of linear partial difference equations (with constant coefficients) on ℤn. Furthermore, we present a method to compute the intersection of an ideal in the algebra of Laurent polynomials with the subalgebra of all polynomials.

The Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant Coefficients

This paper gives a survey of past work in the treated subject and also contains several new results. We solve the Cauchy problem for linear systems of partial difference equations on general integral lattices by means of suitable transfer operators and show that these can be easily computed with the help of standard implementations of Gröbner basis algorithms. The Borel isomorphism permits to transfer these results to systems of partial differential equations. We also solve the Cauchy problem for the function spaces of convergent power series and for entire functions of exponential type. The unique solvability of the Cauchy problem implies that the considered function spaces are large injective cogenerators for which the duality between finitely generated modules and behaviours holds. Already in the beginning of the last century C. Riquier considered and solved problems of the type discussed here.

Gröbner bases with coefficients in rings

This article gives a short introduction to the theory of Grobner bases in a class of rings, which includes rings of differential operators and polynomial rings over commutative noetherian rings. A definition of reduced Grobner bases for these rings is proposed.

Gro¨bner bases with respect to generalized term orders and their application to the modelling problem

Abstract We present an algorithm to decide whether a homogeneous linear partial difference equation with constant coefficients provides an unfalsified model for a finite set of observations, which consist in multiindexed signals, known on a finite subset of N n . To this aim we introduce the concept of “generalized term order” and extend the theory of Grobner bases accordingly.

On certain linear spaces of nilpotent matrices

In this note we describe explicitly those linear spaces of nilpotent matrices, which are stable under conjugation by diagonal matrices. As an application, we deduce a generalized version of a result of Valcher and Fornasini on pairs of nonnegative nilpotent matrices.

Closures of SL(2)-orbits in projective spaces

LetB be a Borel-subgroup ofSL(2,C). In this note we describe the closures of arbitrarySL(2,C)-andB-orbits in the projective space of regularSL(2,C)-modules by means of simple combinatorial data. We give a criterion to detect whether the number of orbits in an orbit-closure is finite or not.

Über gewisseG-stabile Teilmengen in projektiven Räumen

LetG be a connected affine algebraic group over an algebraically closed field of characteristic 0. LetN be a regularG-module andP(N) its projective space. In this article we study those locally closedG-stable subsets ofP(N) which contain in everyG-orbit a fixed point of a maximal unipotent subgroup ofG. Varieties of this type which contain only one closed orbit are classified by “painted monoids”. Necessary and sufficient conditions on a painted monoid are given so that the corresponding variety is smooth.

Normale Einbettungen von Sphärischen Homogenen Räumen

In [LV] wurde eine Methode zur Klassifikation der normalen Einbettungen von homogenen Raumen reduktiver Gruppen entwickelt. Diese Methode soll hier moglichst leicht lesbar dargestellt werden. Zur Vereinfachung beschranken wir uns auf spharische homogene Raume. Beweise werden weggelassen, dafur werden die auftretenden Begriffe genau definiert und durch Beispiele erlautert.

On Invariant Relations between Zeros of Polynomials

ABSTRACT We show that the relation ideal of a univariate polynomial with Galois group G is generated by G-invariant relations; these relations arise from an arbitrary set of generators of the invariant ring of G in a natural way. When G is the symmetric group, the most obvious generators of this kind form an H-basis (in the sense of Macaulay) of the relation ideal. This means that there is an effective way to write any given relation in terms of these generators. We further show that such a result cannot be expected to hold for the alternating group.