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## On the monodromy conjecture for non-degenerate hypersurfaces

arxiv.org.
math.
Cornell University
,
2016.
No. arXiv:1309.0630v4.

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain higher-dimensional analogues of their results, which, in particular, prove the conjecture for all isolated singularities of 4 variables, as well as for many classes of non-isolated and higher-dimensional singularities. One of the tools that we introduce is a certain extension of the notion of a supermodular function, which may be of independent interest in convex analysis and game theory.

Esterov A. I., Compositio Mathematica 2019 Vol. 155 No. 2 P. 229-245

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.
In particular, our result proves the multivariate ...

Added: February 5, 2019

Esterov A. I., Takeuchi K., Lemahieu A., Journal of the European Mathematical Society 2021

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations --- the Denef--Loeser conjecture on topological zeta functions --- is open for surface singularities. We prove it for a wide ...

Added: November 28, 2020

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Lvovsky S., On surfaces with zero vanishing cycles / Cornell University. Series math "arxiv.org". 2013. No. 1305.2205.

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic. ...

Added: October 3, 2013

Vyugin I. V., Левин Р. И., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 326-343

An analog of the classical Riemann-Hilbert problem formulated for classes of difference and q-difference systems is considered. We propose some strengthening of Birkhoff's existence theorem. ...

Added: August 18, 2017

Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597-613

We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...

Added: August 30, 2019

Esterov A. I., Gusev G. G., Mathematische Annalen 2016 Vol. 365 No. 3 P. 1091-1110

We generalize the Abel–Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued ...

Added: February 27, 2017

Serge Lvovski, On monodromy groups of del Pezzo surfaces / Cornell University. Series arXiv "math". 2017.

We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of ...

Added: June 14, 2017

Esterov A. I., Takeuchi K., Ando K., Advances in Mathematics 2015 Vol. 272 P. 1-19

We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson. In particular, we compute the monodromy zeta-function. ...

Added: October 10, 2014

Esterov A. I., Discrete and Computational Geometry 2010 Vol. 44 No. 1 P. 96-148

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving mixed fiber polyhedra and Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and ...

Added: December 10, 2012

Serge Lvovski, Manuscripta Mathematica 2014 Vol. 145 P. 235-242

We show that using an idea from a paper by Van de Ven one may obtain a
simple proof of Zak's classification of smooth projective surfaces
with zero vanishing cycles. This method of proof allows one to extend
Zak's theorem to the case of finite characteristic. ...

Added: October 14, 2014

Brav C. I., Thomas H., Compositio Mathematica 2014 Vol. 150 No. 3 P. 343-333

We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. ...

Added: September 29, 2014

Yu. Burman, Serge Lvovski, Moscow Mathematical Journal 2015 Vol. 15 No. 1 P. 31-48

Suppose that C ⊂ P^2 is a general enough nodal plane curve
of degree > 2, : \hat C → C is its normalization, and π: C′ → P^1 is a finite
morphism simply ramified over the same set of points as a projection
pr_p ◦ν : \hat C → P1, where p ∈ P^2 ...

Added: January 14, 2015

Esterov A. I., Успехи математических наук 2017 Т. 72 № 4 С. 197-198

The degree of the bifurcation locus of a generic polynomial map is computed in terms of Newton polytopes of the components of the map. ...

Added: November 9, 2017

Казарновский Б. Я., Хованский А. Г., Esterov A. I., Успехи математических наук 2021 Т. 76 № 1 С. 95-190

The notions of Newton polytope, toric variety, tropical geometry and Groebner basis established fundamental relations between the algebraic and convex geometries. This survey presents the state of the art of the interfaces between these notions. ...

Added: October 27, 2020

Esterov A. I., Gusev G. G., Journal of symbolic computation 2015 Vol. 68-2 P. 116-130

We classify general systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorithm to evaluate the single solution of such a system. ...

Added: October 24, 2013

V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Akhtar M., Coates T., Galkin S. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 094 P. 1-707

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with ...

Added: September 14, 2013

Gusein-Zade S., Journal of Algebra and its Applications 2018 Vol. 17 No. 10 P. 1-13

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring Bˆ(G) of a finite group G is defined. An ...

Added: October 27, 2020

Vyugin I. V., Гонцов Р. Р., Успехи математических наук 2012 Т. 67 № 3 (405) С. 183-184

Получено обобщение результата Ильяшенко-Хованского, утверждающего, что разрешимость в квадратурах фуксовой системы с малыми коэффициентами эквивалентна ее треугольности. В работе этот результат обобщен на случай систем с малыми собственными значениями матриц вычетов. ...

Added: February 21, 2013

Lvovsky S., On monodromy in families of elliptic curves over C / Cornell University. Series arXiv "math". 2018.

We show that if we are given a smooth non-isotrivial family of elliptic curves over ℂ with a smooth base B for which the general fiber of the mapping J:B→𝔸^1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1(⋅,ℤ) of the fibers) coincides with SL(2,ℤ); if the general fiber has m≥2 connected components, then the ...

Added: December 5, 2018

Бухштабер В. М., Glutsyuk A., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 62-104

Abstract—We study a family of double confluent Heun equations of the form LE = 0, where
L = L(λ,μ,n) is a family of second-order differential operators acting on germs of holomorphic
functions of one complex variable. They depend on complex parameters λ, μ, and n. The
restriction of the family to real parameters satisfying the inequality λ + μ^2>0 ...

Added: June 29, 2018

Kiritchenko V., Smirnov E., Timorin V., Snapshots of modern mathematics from Oberwolfach (Germany) 2015

In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion. ...

Added: July 10, 2015

Bruno A., Parusnikova A., Доклады Академии наук 2012 Т. 442 № 5 С. 583-588

В работе методами степенной геометрии найдены все асимптотические разложения решений пятого уравнения Пенлеве в окрестности его не особой точки для всех значений четырех комплексных параметров уравнения. Получено 10 семейств разложений решений уравнения, одно из которых не было известно раньше. Три разложения являются рядами Лорана, а остальные семь – рядами Тейлора. Все они сходятся в (проколотой) ...

Added: November 30, 2012