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## Newton polyhedra of discriminants of projections

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 generalizes a number of similar known results.

Takeuchi K., Esterov A. I., Lemahieu A., On the monodromy conjecture for non-degenerate hypersurfaces / 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

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

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

Kiritchenko V., Timorin V., Smirnov E., Oberwolfach Reports 2011 Vol. 8 No. 3 P. 2341-2344

We construct generalized Newton polytopes for Schubert subvarieties in the variety of complete flags in C^n . Every such “polytope” is a union of faces of a Gelfand–Zetlin polytope (the latter is a well-known Newton–Okounkov body for the flag variety). These unions of faces are responsible for Demazure characters of Schubert varieties and were originally used ...

Added: November 17, 2012

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., 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., Advances in Mathematics 2013 Vol. 245 P. 534-572

Which polynomial in the coefficients of a system of algebraic equations should be called its discriminant? We prove a package of facts that provide a possible answer. Let us call a system typical, if the homeomorphic type of its set of solutions does not change as we perturb its (non-zero) coefficients. The set of all ...

Added: May 15, 2013

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

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

Providence : American Mathematical Society, 2012

The volume is to contain the proceedings of the 13th conference AGCT as well as the proceedings of the conference Geocrypt. The conferences focus on various aspects of arithmetic and algebraic geometry, number theory, coding theory and cryptography. The main topics discussed at conferences include the theory of curves over finite fields, theory of abelian ...

Added: January 3, 2013

Feigin B. L., Buryak A., Journal of Geometry and Physics 2012 Vol. 62 No. 7 P. 1652-1664

The moduli space M(r,n) of framed torsion free sheaves on the projective plane with rank r and second Chern class equal to n has the natural action of the (r+2)-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case ...

Added: September 20, 2012

Trubochkina N. K., Качество. Инновации. Образование 2012 Т. 84 № 5 С. 76-82

Scientific and exploratory research and mathematical simulation of a fractal designresults are presented. Decorative fractal patterns for textile and construction industries (models, murals, stained glass) have been developed. Fractals, which can be attributed to a class of fractal artdeveloped. Technology of frost-and water-resistant seamless large area fractal frescoes developed. Technology cost much less than for ...

Added: November 21, 2012

Feigin M., Shramov K., International Mathematics Research Notes 2012 Vol. 2012 No. 15 P. 3375-3414

We consider representations of rational Cherednik algebras that are particular ideals in the ring of polynomials. We investigate convergence of the integrals that express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types B and D for the singular values suggested by Cherednik with at ...

Added: September 13, 2012

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

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

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 2 P. 337-361

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gln), and formulate a conjectural answer for the small quantum cohomology rings in terms of ...

Added: October 9, 2012

Prokhorov Y., Journal of Algebraic Geometry 2012 Vol. 21 No. 3 P. 563-600

We classify all finite simple subgroups of the Cremona group Cr3(C). ...

Added: September 19, 2012

Popov V., Problems for the problem session / Centro Internazionale per la Ricerca Matematica. Series CIRM "Electronic preprint server". 2012. No. нет.

Some problems on the structure of the Cremona groups formulated (with comments) by the author at the International conference Birational and Affine Geometry, Levico Terme (Trento), 29.10.12--03.11.12 ...

Added: January 9, 2013

Feigin E., Cerulli Irelli G., Reineke M., Algebra & Number Theory 2012 Vol. 6 No. 1 P. 165-194

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and ...

Added: June 29, 2012

Andrey Soldatenkov, International Mathematics Research Notices 2012 Vol. 2012 No. 15 P. 3483-3497

A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in GL(n, H). There is a well-known construction of hypercomplex structures on Lie ...

Added: January 17, 2013

Feigin E., Selecta Mathematica, New Series 2012 Vol. 18 No. 3 P. 513-537

Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains ...

Added: August 31, 2012

Samovol V. S., Математические заметки 2014 Т. 95 № 6 С. 911-925

The asymptotic properties of nonoscillating solutions of Emden-Fowler-type equations of arbitrary order are considered. The paper contains the results of the study of the asymptotic properties of solutions with integer-valued asymptotics as well as of solutions arising from the rapid decrease of the coefficient of the equation. To analyze the asymptotic behavior of solutions of ...

Added: February 13, 2014

V.S.Samovol, Mathematical notes 2015 Vol. 97 No. 1-2 P. 100-110

Emden-Fowler type equations of arbitrary order are considered. The paper contains asymptotic estimates of nonoscillating continuable and noncontinuable solutions of such equations. ...

Added: October 8, 2015

M. : Higher School of Economics Publishing House, 2012

Toric geometry exhibited a profound relation between algebra and topology on one side and combinatorics and convex geometry on the other side. In the last decades, the interplay between algebraic and convex geometry has been explored and used successfully in a much more general setting: first, for varieties with an algebraic group action (such as ...

Added: November 17, 2012