Uli Walther

Homological methods for hypergeometric families

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(\beta) arising from a d x n integer matrix A and a parameter \beta \in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter \beta is rank-jumping for H_A(\beta) if and only if \beta lies in the Zariski closure of the set of \ZZ^d-graded degrees \alpha where the local cohomology \bigoplus_{i<d}H^i_\frakm(\CC[\NN A])_\alpha of the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(\beta) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d.

Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties

In this paper we present algorithms that compute certain local cohomology modules associated to ideals in a ring of polynomials containing the rational numbers. In particular, we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. Our approach is based on the theory of D-modules.

Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements

Let Q ∈ C[x1, . . . , xn] be a homogeneous polynomial of degree k > 0. We establish a connection between the Bernstein-Sato polynomial bQ(s) and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integer uQ = max{i ∈ Z : bQ(−(i+n)/k) = 0} bounds the top degree (as differential form) of the elements in H DR (Q(1), C). The link is provided by the relative de Rham complex and D-module algorithms for computing integration functors. As an application we determine the Bernstein-Sato polynomial bQ(s) of a generic central arrangement Q = ∏k i=1 Hi of hyperplanes. We obtain in turn information about the cohomology of the Milnor fiber of such arrangements related to results of Orlik and Randell who investigated the monodromy. We also introduce certain subschemes of the arrangement determined by the roots of bQ(s). They appear to correspond to iterated singular loci.

On the computation of switching surfaces in optimal control: a Grobner basis approach

A number of problems in control can be reduced to finding suitable real solutions of algebraic equations. In particular, such a problem arises in the context of switching surfaces in optimal control. A powerful new methodology for doing symbolic manipulations with polynomial data has been developed and tested, namely the use of Grobner bases. We apply the Grobner basis technique to find effective solutions to the classical problem of time-optimal control.

On the Lyubeznik numbers of a local ring

We collect some information about the invariants λp,i(A) of a commutative local ringA containing a field introduced by G. Lyubeznik in 1993 (Finiteness properties of local cohomology modules, Invent. Math. 113, 41– 55). We treat the cases dim(A) equal to zero, one and two, thereby answering in the negative a question raised in Lyubeznik’s paper. In fact, we will show that λp,i(A) has in the two-dimensional case a topological interpretation.

A localization algorithm for D -modules

Abstract We present a method to compute the holonomic extension of a D -module from a Zariski open set in affine space to the whole space. A particular application is the localization of coherent D -modules which are holonomic on the complement of an affine variety.

Local cohomology modules supported at determinantal ideals

We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer coefficients---supported at generic determinantal ideals---and also obtain results on F-modules and D-modules that are likely to be of independent interest.

Computing Homomorphisms Between HolonomicD-modules

Let K?C be a subfield of the complex numbers, and let D be the ring of K -linear differential operators on R=Kx1, . . . , xn. If M and N are holonomic left D -modules we present an algorithm that computes explicit generators for the finite dimensional vector spaceHomD(M, N). This enables us to answer algorithmically whether two given holonomic modules are isomorphic. More generally, our algorithm can be used to get explicit generators forExtDi(M, N) for any i in the sense of Yoneda.

Bockstein homomorphisms in local cohomology

Abstract Let R be a polynomial ring in finitely many variables over the integers. Fix an ideal 𝔞 of R. We prove that for all but finitely many prime integers p, the Bockstein homomorphisms on local cohomology, , are zero. This vanishing of Bockstein homomorphisms is predicted by Lyubezniks conjecture which states that when R is a regular ring, the modules have finitely many associated prime ideals. We further show that when R is replaced by a hypersurface ring, the vanishing of Bockstein homomorphisms—as well as the analogue of Lyubezniks conjecture—need not hold. Bockstein homomorphisms have their origins in algebraic topology; we establish a connection between algebraic and topological Bockstein homomorphisms through Stanley–Reisner theory.

Computing the cup product structure for complements of complex affine varieties

Abstract Let X= C n . In this paper we present an algorithm that computes the cup product structure for the de Rham cohomology ring H dR • (U; C ) where U is the complement of an arbitrary Zariski-closed set Y in X . Our method relies on the fact that Tor is a balanced functor, a property which we make algorithmic, as well as a technique to extract explicit representatives of cohomology classes in a restriction or integration complex. We also present an alternative approach to computing V -strict resolutions of complexes that is seemingly much more efficient than the algorithm presented in Walther (J. Symbolic Comput. 29 (2000) 795–839). All presented algorithms are based on Grobner basis computations in the Weyl algebra.