arXiv subjects
Publications and source records attributed to Robin van der Veer.
In this paper, we review several results on the zero loci of Bernstein-Sato ideals related to singularities of hypersurfaces. This is an exposition for the Frontiers of Science Awards in Mathematics presenting results from one of our articles, with history, motivation, and further developments.
We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more generally for semi-quasihomogeneous hypersurfaces. We also give a strange generalization allowing a twist by certain differential forms.
We give estimates for the zero loci of Bernstein-Sato ideals. An upper bound is proved as a multivariate generalisation of the upper bound by Lichtin for the roots of Bernstein-Sato polynomials. The lower bounds generalise the fact that log-canonical thresholds, small jumping numbers of multiplier ideals, and their real versions provide roots of Bernstein-Sato polynomials.
In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein--Sato theory. We investigate the critical points of very affine varieties and study their asymptotic behavior. We relate these asymptotics to particular rays in the tropical variety as well as to Bernstein--Sato ideals and give a connection to Maximum Likelihood Estimation in Statistics.
A log generic hypersurface in $\mathbb{P}^n$ with respect to a birational modification of $\mathbb{P}^n$ is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product $f=f_1\ldots f_p$ of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for the tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log very-generic.
We have recently proved a precise relation between Bernstein-Sato ideals of collections of polynomials and monodromy of generalized nearby cycles. In this article we extend this result to other ideals of Bernstein-Sato type.
In this article we investigate the fibers of relative $D$-modules. In general we prove that there exists an open, Zariski dense subset of the vanishing set of the annihilator over which the fibers of a cyclic relative $D$-module are non-zero. Next we restrict our attention to relatively holonomic $D$-modules. For this class we prove that the fiber over every point in the vanishing set of the annihilator is non-zero. As a consequence we obtain new proofs of a conjecture of Budur which was recently proven by Budur, van der Veer, Wu and Zhou, as well as a new proof of a theorem of Maisonobe. Moreover, we also obtain a diagonal specialization result for Bernstein-Sato ideals.
We prove a conjecture of the first author relating the Bernstein-Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology.
In this article we introduce a new matroid invariant, a combinatorial analog of the topological zeta function of a polynomial. More specifically we associate to any ranked, atomic meet-semilattice L a rational function Z(L,s), in such a way that when L is the lattice of flats of a complex hyperplane arrangement we recover the usual topological zeta function. The definition is in terms of a choice of a combinatorial analog of resolution of singularities, and the main result is that Z(L,s) does not depend on this choice and depends only on L. Known properties of the topological zeta function provide a source of potential complex realisability test for matroids.