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Lacunas and local algebraicity of volume functions
Journal of Singularities. 2018. Vol. 18. P. 350-357.
The volume cut off by a hyperplane from a bounded body in 2k-dimensional space never is an algebraic function on the space of hyperplanes: for k=1 it is the famous Lemma XXVIII from Newton's Principia. Following an analogy of these volume functions with the solutions of hyperbolic PDE's, we study the local version of the same problem: can such a volume function coincide with an algebraic one at least in some domains of the space of hyperplanes, intersecting the body? We prove some homological and geometric obstructions to this integrability property. Based on these restrictions, we find a family of examples of such "locally integrable" bodies in Euclidean spaces.
V.A.Vassiliev, Arnold Mathematical Journal 2020 Vol. 6 No. 2 P. 291-309
V. Arnold’s problem 1987–14 from his Problems book asks whether there exist bodies with smooth boundaries in R^N (other than the ellipsoids in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body depends algebraically on the hyperplane. We present a series of very realistic candidates for the role ...
Added: August 17, 2020
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
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
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
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
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
Lvovski Serge, / Cornell University. Series math "arxiv.org". 2021.
We construct a large class of projective threefolds with one node (aka non-degenerate quadratic singularity) such that their small resolutions are not projective. ...
Added: October 28, 2021
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
Alexander Esterov, Lang L., Geometry and Topology 2021 Vol. 25 No. 6 P. 3053-3077
Let C_d be the space of non-singular, univariate polynomials of degree d. The Viète map V sends a polynomial to its unordered set of roots. It is a classical fact that the induced map V_∗ at the level of fundamental groups realises an isomorphism between π_1(C_d) and the Artin braid group B_d. For fewnomials, or equivalently for the intersection C of C_d with a collection of coordinate ...
Added: October 27, 2020
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
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
Lvovsky S., / 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
Piontkovski D., La Scala R., Tiwari S., / Cornell University. Series math "arxiv.org". 2019.
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class ...
Added: October 7, 2019
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
Serge Lvovski, / 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
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
V.A.Vasil'ev, Mathematical notes 2019 Vol. 106 No. 6 P. 894-898
Any compact body with regular boundary in R^N defines a two-valued function on the
space of affine hyperplanes: the volumes of the two parts into which these hyperplanes cut the body.
This function is never algebraic if N is even and is very rarely algebraic if N is odd: all known bodies
defining algebraic volume functions are ellipsoids ...
Added: December 6, 2019
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
Кузютин Д. В., Смирнова Е. Л., Razgulyaeva L. et al., Изд-во МБИ, 2010
The present textbook is intended for students preparing to study mathematics at a higher education institution, to prepare to pass the exam. ...
Added: November 4, 2014
12890250, Tulyakov D., Aptekarev A. I. et al., Journal of Computational and Applied Mathematics 2009 Vol. 233 No. 3 P. 602-616
We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the HermitePadé approximants. Solutions of such problems can be expressed bymeans of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic ...
Added: January 25, 2013
Verbitsky M., Mehrotra S., Markman E., European Journal of Mathematics 2019 Vol. 5 No. 3 P. 964-1012
Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over 𝑀×𝑀 there exists a rank 2𝑛−2 reflexive hyperholomorphic sheaf 𝐸_𝑀, whose fiber over a non-diagonal point (𝐹_1, 𝐹_2) is Ext^1_𝑆 (𝐹_1, 𝐹_2). The sheaf 𝐸_𝑀 can be deformed along some twistor path to a sheaf 𝐸_𝑋 over the Cartesian square 𝑋×𝑋 of every Kähler manifold X deformation equivalent to M. We prove that 𝐸_𝑋 is ...
Added: March 11, 2019
Takeuchi K., Esterov A. I., Lemahieu A., / Cornell University. Series math "arxiv.org". 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 ...
Added: September 18, 2017
Lvovsky S., / 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