arXiv subjects

Thomas Bitoun

Publications and source records attributed to Thomas Bitoun.

10 recordsLinked to original sources

arXiv2024

Bernstein-Sato theory modulo $p^m$

For fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z} / p^m$-coefficients, compatible with existing theory in the case $m = 1$. We show that the ``roots" of such polynomials are rational and we show that the negative roots agree with those of the mod-$p$ reduction. We give examples to show that, surprisingly, roots may be positive in this context. Moreover, our construction allows us to define a notion of ``strength" for roots by measuring $p$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$p$ reduction.

math.AC↗

arXiv2023

On the D-module of an isolated singularity

Let Z be the germ of a complex hypersurface isolated singularity of equation f, with Z at least of dimension 2. We consider the family of analytic D-modules generated by the powers of 1/f and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of {f=0} in the neighborhood of the singularity.

math.AG↗

arXiv2022

On centralizers in Azumaya domains

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo $p.$

math.RA↗

arXiv2018

Extending meromorphic connections to coadmissible D-cap-modules

We investigate when a meromorphic connection on a smooth rigid analytic variety $X$ gives rise to a coadmissible $\mathcal{D}_X$-cap-module, and show that this is always the case when the roots of the corresponding $b$-functions are all of positive type. On the other hand, we also give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.

math.AG↗

arXiv2018

The number of master integrals as Euler characteristic

We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky polynomial $\mathcal{G}$. By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial $\mathcal{G}$.

hep-th↗

arXiv2017

Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.

hep-th↗

arXiv2017

Length of local cohomology in positive characteristic and ordinarity

Let $D$ be the ring of Grothendieck differential operators of the ring $R$ of polynomials in $d\geq3$ variables with coefficients in a perfect field of positive characteristic $p.$ We compute the $D$-module length of the first local cohomology module $H^1_f(R)$ of $R$ with respect to an irreducible polynomial $f$ with an isolated singularity, for $p$ large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the structure sheaf of the exceptional divisor of a resolution of the singularity. Our proof rests on a tight closure computation due to Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero $\mathcal{D}$-module whose length is expected to equal that above, for ordinary primes.

math.AG↗

arXiv2016

On D-modules related to the b-function and Hamiltonian flow

Let f be a quasi-homogeneous polynomial with an isolated singularity. We compute the length of the D-modules $Df^c/Df^{c+1}$ generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. For 1/f we obtain one more than the reduced genus of the singularity. We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the aforementioned quotient is nonzero when c is a root of the b-function of f (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these D-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.

math.AG↗

arXiv2014

On a theory of the $b$-function in positive characteristic

We present a theory of the $b$-function (or Bernstein-Sato polynomial) in positive characteristic. Let $f$ be a non-constant polynomial with coefficients in a perfect field $k$ of characteristic $p>0.$ Its $b$-function $b_f$ is defined to be an ideal of the algebra of continuous $k$-valued functions on $\mathbb{Z}_p.$ The zero-locus of the $b$-function is thus naturally interpreted as a subset of $\mathbb{Z}_p,$ which we call the set of roots of $b_f.$ We prove that $b_f$ has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a} and is in terms of $D$-modules, where $D$ is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of $b_f$ and relate it to the test ideals of $f.$

math.AG↗

arXiv2010

On the $p$-supports of a holonomic $\mathcal{D}$-module

For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y'},$ the cotangent space of the Frobenius twist $Y'$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T^*_{Y'},$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T^*_{Y'}.$ We study here the family of $p$-supports assigned to the reductions modulo primes $p$ of a holonomic $\mathcal{D}$-module. We prove that the Azumaya algebra of differential operators splits on the regular locus of the $p$-support and that the $p$-support is a Lagrangian subvariety of the cotangent space, for $p$ large enough. The latter was conjectured by Kontsevich. Our approach also provides a new proof of the involutivity of the singular support of a holonomic $\mathcal{D}$-module, by reduction modulo $p.$

math.AG↗