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Polynomial Equations in Subgroups and Applications
P. 273–297.
In book
Vyugin I. V., Konyagin S. V., Shparlinski I. Switzerland: Springer, 2022.
Aleshina S., 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
Vyugin I. V., Konyagin S. V., Shparlinski I., Switzerland: Springer, 2022.
Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples ...
Added: January 21, 2023
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
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
Макарычев С. В., Vyugin I. V., Arnold Mathematical Journal 2019 Vol. 5 No. 1 P. 105–121
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 ...
Added: August 13, 2019
Shvedov A. S., Проблемы управления 2018 № 1 С. 21–29
This paper surveys some parts of approximation theory for functions of one and several real variables. Approximation of functions by algebraic polynomials is a classical theory. The paper contains some results from this theory. First results on approximation of functions by neural networks and fuzzy systems have appeared as responses on practical requirements. It was ...
Added: February 17, 2018
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
Kochetkov Y., Функциональный анализ и его приложения 2002 Т. 36 № 3 С. 83–87
Some special polynomial systems (antivandermond systems) and definition fields of their solutions are studied. In the case of four variables it is proved that definition field is an extension of degree 12 of real quadratic field. ...
Added: June 28, 2012