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

We present a theory of the b-function (or Bernstein–Sato polynomial) in positive characteristic. Let f be non-constant polynomial with coefficients perfect field k characteristic $$p>0.$$ Its $$b_f$$ is defined to an ideal algebra continuous k-valued functions on $${\mathbb {Z}}_p.$$ The zero-locus thus naturally interpreted as subset {Z}}_p,$$ which we call set roots $$b_f.$$ prove that has finitely many and they are negative rational numbers. Our construction builds earlier work Mustaţă...

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

Let $f$ be a quasi-homogeneous polynomial with an isolated singularity in $\mathbf{C}^{n}$ . We compute the length of ${\mathcal{D}}$ -modules ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers terms Hodge filtration on top cohomology Milnor fiber. When $\unicode[STIX]{x1D706}=-1$ we obtain one more than reduced genus ( $\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for $Z$ exceptional fiber resolution singularities). conjecture that this...

Abstract Let D be the ring of Grothendieck differential operators R polynomials in d ≥ 3 variables with coefficients a perfect field characteristic p. We compute D-module length 1st local cohomology module ${H^{1}_{f}}(R)$ respect to polynomial f an isolated singularity, for p large enough. The expression we give is terms Frobenius action on top coherent exceptional fibre resolution singularity. Our proof rests tight closure computation Hara. Since above quite different from that...

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

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

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

Abstract We investigate when a meromorphic connection on smooth rigid analytic variety 𝑋 gives rise to coadmissible <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:munder> <m:mover accent="true"> <m:mi mathvariant="script">D</m:mi> <m:mo movablelimits="false">⏜</m:mo> </m:mover> <m:mi>X</m:mi> </m:munder> </m:math> \overparen{\mathcal{D}}_{X} -module, and show that this is always the case roots of corresponding 𝑏-functions are all positive type. also use theory give an example...

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

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

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

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

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

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