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Solutions of Polynomial Equations in Subgroups of F_p^*
Arnold Mathematical Journal. 2019. Vol. 5. No. 1. P. 105-121.
Макарычев С. В., Vyugin I. V.
We present an upper bound on the number of solutions of an algebraic equation P(x,y)=0 where x and y belong to the union of cosets of some subgroup of the multiplicative group κ∗ of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927–1942, 2013) to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy.
Vyugin I. V., Математические заметки 2019 Т. 106 № 2 С. 212-221
An upper bound for the number of field elements that can be taken to roots of unity of fixed multiplicity by means of several given polynomials is obtained. This bound generalizes the bound obtained by V'yugin and Shkredov in 2012 to the case of polynomials of degree higher than 1. This bound was obtained both over ...
Added: October 9, 2019
В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2016 Т. 468 № 5 С. 499-501
A general theorem on the purity of of invariant field extensions
is proved. Using it, a criterion of rational triangulability of connected solvable
affine algebraic subgroups of the Cremona groups is obtained. This criterion is
applied for proving the existence of rationally nontriangulable subgroups of the
above form and for proving their stable rational triangulability. The latter property
answers in ...
Added: May 16, 2016
Vyugin I. V., Солодкова Е. В., Шкредов И. Д., Математические заметки 2016 Т. 100 № 2 С. 185-195
By means Stepanov's method the bound of cardinality of the intersection of additive shifts of several subgroups of multiplicative group of the finite field was obtained. This bound apply to some question of additive decomposition of subgroups. ...
Added: January 29, 2016
Vyugin I. V., Макарычев С. В., Arnold Mathematical Journal 2019
In this paper we present a new bound of the number of solutions of a polynomial equation $P(x,y)=0$ over the field $\overline{\mathbb{F}}_p$ that belong to a union of cosets $(x,y)\in\bigcup_{i=1}^hg_i'G\times g_i''G$.
This bound improves a case of the bound of Corvaja and Zannier (see \cite{C-Z}) in the case $h\gg 1$. Using this bound we obtain estimates ...
Added: December 14, 2017
Nabebin A. A., Tarasikov A. S., M. : -, 2012
The manual contains necessary data from homogeneous and inhomogeneous universal algebras, systems of axioms for the basic algebraic structures (arithmetics, monoids, subgroups, groups, partial ordering, rings, fields). It is described axiomatic programming language OBJ3 with examples of programs in this language. It is intended to students of universities of specialities: the applied mathematics, informatics, computer ...
Added: March 13, 2013
Алешина С. А., Vyugin I. V., Математические заметки 2023 Т. 113 № 1 С. 3-10
We generalize two results in the papers [1:x003] and [2:x003] about sums of subsets of Fp to the more general case in which the sum x+y is replaced by P(x,y), where P is a rather general polynomial. In particular, a lower bound is obtained for the cardinality of the range of P(x,y), where the variables x and y belong to a subgroup G of the multiplicative group of the field Fp. We also prove that ...
Added: January 21, 2023
Popov V., / 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
Kang M., Yuri Prokhorov, Journal of Algebra 2010 Vol. 324 No. 9 P. 2166-2197
Added: December 5, 2013
Vyugin I. V., Макарычев С. В., Shparlinski I. et al., Quarterly Journal of Mathematics 2020 Vol. 71 No. 2 P. 637-648
We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the ‘giant component’ and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo p. This is a step towards the conjecture that there are no such nodes at ...
Added: September 12, 2020
Kosov E., International Mathematics Research Notices 2021 Vol. 2021 No. 21 P. 16492-16508
This paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the L^2-distance between these polynomials. ...
Added: July 31, 2020
Kulikov V. S., Математические заметки 2012 Т. 91 № 4 С. 547-550
Мы изучаем отображение, переводящее приведенный комплексный много член 𝑓(𝑥) степени 𝑛 без кратных корней в набор из 𝑛 значений его производной
в его корнях. Дается ответ на вопрос, поставленный Ю.С. Ильяшенко.
Библиография: 4 названия. ...
Added: February 6, 2013
Vyugin I. V., Макарычев С. В., / Cornell University. Series arXiv "math". 2015.
We present an upeer bound of the number of solutions (x,y) of a polynomial equation P(x,y)=0 over a field F_p in the case where x,y from G, G is a subgroup of F_p^*. ...
Added: June 22, 2015
V. L. Popov, Doklady Mathematics 2016 Vol. 93 No. 3 P. 307-309
A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The ...
Added: June 29, 2016
Vyugin I. V., Shkredov I. D., Solodkova E. V., / Cornell University. Series math "arxiv.org". 2014.
The paper is devoted to some applications of Stepanov method. In the first part of the paper we obtain the estimate of the cardinality of the set, which is obtained as an intersection of additive shifts of some different subgroups of F_p^*. In the second part we prove a new upper bound for Heilbron's exponential ...
Added: September 2, 2014
V. L. Popov, Zarhin Y. G., Doklady Mathematics 2018 Vol. 98 No. 3 P. 600-602
We classify the types of root systems R in the rings of integers of number fields K such that the Weyl
group W(R) lies in the group generated by Aut(K) and multiplications by the elements of K*. ...
Added: December 26, 2018
I. V. V’yugin, Journal of Mathematical Sciences 2023 Vol. 270 P. 665-673
We estimate the orders of zeros of polynomials f(x) = P(u1(x), u2(x), . . . , un(x)) in the fundamental system of solutions to a linear Fuchsian differential equation. We introduce the notions of A- and (∞, A)-algebraic independence and prove that the system of functions xt, u1(x), u2(x), . . . , un(x) is ...
Added: March 5, 2024
Blokh A., Oversteegen L., Ptacek R. M., / Cornell University. Series arXiv "math". 2015.
We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations ∼ on the unit circle invariant under σ2). To each lamination we associate a particular {\em geolamination} (the collection L∼ of points of the circle and edges of convex hulls of ∼-equivalence classes) so that the closure of the set of all of them is a compact metric ...
Added: July 7, 2015
Ulyanov V.V., Theory Probability and its Applications 2016 Vol. 60 No. 2 P. 325-336
This paper deals with different properties of polynomials in random elements: bounds for characteristic functionals of polynomials, a stochastic generalization of the Vinogradov mean value theorem, the characterization problem, and bounds for probabilities to hit the balls. These results cover the cases when the random elements take values in finite as well as infinite dimensional ...
Added: March 12, 2017
Vyugin I. V., Шкредов И. Д., Математический сборник 2012 Т. 203 № 6 С. 81-100
В работе получена оценка мощности множества, полученного пересечением подгруппы мультипликативной группы поля Z_p и нескольких ее аддитивных сдвигов. ...
Added: February 21, 2013
Shtern A., Математика в высшем образовании 2007 № 5 С. 95-100
Излагается программа курса лекций “Многочлены от нескольких переменных (элементарное введение в коммутативную алгебру)”. Обсуждаются методические особенности курса и его место в системе дополнительного математического образования. В контексте постановки вопроса о дополнительном образовании анализируются связи предлагаемого курса с некоторыми вопросами, традиционно входящими в программы кружковой работы со школьниками. ...
Added: December 17, 2018
Bogachev V., Успехи математических наук 2016 Т. 71 № 4 С. 107-154
We give a survey of results about distributions of polynomials on multidimensional spaces with measures. ...
Added: February 1, 2017
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021
В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80
Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...
Added: May 3, 2017