Bernstein-Sato ideals

This is a brief survey of recent works by Neil Trudinger and myself on the Bernstein problem and Plateau problem for affine maximal hypersurfaces.

Let $f_1$ and $f_2$ be two semi-universal deformations of quasi homogeneous polynomials in two variables respectively for the weight vectors $\rho_1$ and $\rho_2$ such that they satisfy similar conditions to that of semi quasi homogeneous singularities for one weight. By methods inspired by H. Maynadier's, we give an explicit formula for a Bernstein-Sato polynomial involving two affine forms $\rho_i(f_1) s_1 + \rho_i(f_2) s_2 +k$, $i=1,2$. In the particular case $(f_1, f_2)=(...

The classification of isoparametric hypersurfaces with four principal curvatures in the sphere interplays in a deep fashion with commutative algebra, whose abstract and comprehensive nature might obscure a differential geometer's insight into the classification problem that encompasses a wide spectrum of geometry and topology. In this paper, we make an effort to bridge the gap by walking through the important part of commutative algebra central to the classification of such h...

We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.

After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial varieties), we shall proceed to computational applications of resolution of singularities in singularity theory and algebraic geometry, also touching on relations to algebraic statistics and machine learning. Namely, we explain how to compute t...

We prove that in regular $F$-finite rings of positive characteristic, the Bernstein-Sato root set of the tensor product of ideals is the union of their respective Bernstein-Sato root sets. Moreover, by computing some special $F$-thresholds, we provide an explicit description about the Bernstein-Sato roots of a weighted homogeneous polynomial with an isolated singularity at the origin in positive characteristic.

We define the Bernstein-Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.

The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the Vandermonde determinant $\xi$. We use a result of Opdam to produce a lower bound for the $b$-function of $\xi$. This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler for the case of finite Coxeter hyperplane arrangements, proving t...

We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy corollary of known results. We collect these results, we precise some proofs and add few results to get a coherent and complete survey.

We generalize Griffiths' theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface, using the local Bernstein-Sato polynomials, the V-filtration of Kashiwara and Malgrange along the hypersurface and the Brieskorn module of the global defining equation of the hypersurface.