Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/dfc/2012/reports/Vyugin_Report2015.pdf Дата изменения: Sun Dec 27 14:37:23 2015 Дата индексирования: Sun Apr 10 15:23:32 2016 Кодировка: Поисковые слова: arp 220

REPORT ON THE DYNASTY FOUNDATION 2015
ILYA VYUGIN

Results of the 2015 year Results in additive combinatorics. We study an algebraic equation P (x, y ) = 0 over a field Fp , where p is a prime. Let P Fp [x, y ] be a polynomial of two variables x and y , G be a subgroup of F . p We study the upper bound of the number solutions of the polynomial equation, such that x g1 G, y g2 G. The estimate #{(x, y ) | P (x, y ) = 0, x g1 G, y g2 G} 16mn2 (m + n)|G|
2/ 3

.

is obtained using Stepanov method. This estimate was obtained by a different method in the paper [1]. We improve this estimate in average. Let us consider a homogeneous polynomial P (x, y ) of degree n such that deg P (x, 0) 1, P (0, 0) = 0 and l1 , . . . , lh belong to different cosets gi G. We estimate the sum Nh of numbers of solutions of the set of equations: P (x, y ) = li , i = 1, . . . , h,
3 /4 5

x g1 G1 , y g2 G.
2 /3

Then the sum Nh does not exceed 32h n |G| . Now let us consider some generalization of the additive energy which we call polynomial energy. Polynomial energy is the following
q EP (A) = #{(x1 , y2 , x2 , y2 ) | P (x1 , y1 ) = P (x2 , y2 ), x1 , y1 , x2 , y2 A},

( ) 12 Theorem Let us suppose that 100n3 < |G| < p 17 , P Fp [x, y ] is a homoge3 7q +16 q neous polynomial. Then the fol lowing holds: if q 3 then EP (G) C (n, q )|G| 12 ; 2q 4 if q = 4 then EP C (n, q )|G|1+ 3 ln |G|; 2q q if q 5 then EP (G) C (n, q )|G|1+ 3 , where C (n, q ) depends only on n and q . The results of the talk can be found in the paper [2]. Results in analytic theory Consider the system (1) Y (z + 1) = A(z )Y (z ) of linear difference equations, with a rational coefficient n в n-matrix function A(z ) = Ar z r + . . . + A0 . Birkhoff proved that the class of meromorphic equiva( s) lence of such systems is described by a set of characteristic constants {i }, {ckl }. Birkhoff also studied the inverse problem (Generalized Riemann­Hilbert problem for difference equations). We improve the Birkhoff 's result of inverse problem. We prove that for any set of eigenvalues 1 , . . . , n of the matrix Ar , such that i /j R for i = j , there are / matrices A0 , . . . , Ar-1 , such that the system (??) with Ar = diag(1 , . . . , n ) has ( s) the given characteristic constants {dk }, {ckl }. Birkhoff has proved that there are (s) the system with constants {dk + lk }, {ckl }, where lk are some integers.
1

where P (x, y ) Fp [x, y ] is a polynomial.


2

ILYA VYUGIN

3 . , : . : 1. - . 2. . - . . , . , .. . P(x,y)=0, Fp . . , . 2011-2012 , . , , Arnold Mathematical Journal, 3 , . Pap ers [1] (With R.R. Gontsov) Solvability of linear differential systems in the Liouvillian sense // Arnold mathematical journal , Volume 1, Issue 4, P. 445-471, 2015, The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small. [2] (With I.D. Shkredov, E.V. Solodkova) Intersections of multiplicative subgroups // Mathematical notes, 2016, (to be appear). 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 Fp . [3] (With S. Makarychev) The number of solutions of polynomial equation over the field Fp and new bounds of additive energy // Proceedings of the Conference Diferential Equations and Related Topics, 2015, p. 24-26. An algebraic equation P (x, y ) = 0 over a field Fp , where p is a prime is studied. Let G be a subgroup of F . We study the upper bound of the number solutions of p the polynomial equation, such that x g1 G, y g2 G. The estimate #{(x, y ) | P (x, y ) = 0, x g1 G, y g2 G} 16mn2 (m + n)|G|
2/ 3

.

is obtained using Stepanov method. We improve this estimate in average. (The extended version of this paper has been submited to journal.)


REPORT ON THE DYNASTY FOUNDATION 2015

3

Scientific conferences and seminar talks Talks: 1. Conference Diferential Equations and Related Topics in Zaraisk, talk: "The number of solutions of polynomial equation over the field Fp and new bounds of additive energy June 11-12, 2015. Seminar talks: 2. Analitic theory of differential equations in Steklov Mathematical Institute several talks. 3. Seminar on Complex Analysis (Gonchar Seminar) in Steklov Mathematical Institute, talk: "On the generalized Riemann"Hilbert problem for linear difference equations". 4. Seminar Contemporary Problems in Number Theory, talk: "On the number of solutions of polynomial equation P (x, y ) = 0 over Fp , such that x and y belong to some subgroup of F ". p Teaching [1] Analitic theory of differential equations (lectures) Special course in HSE and IUM (joint with V.A. Poberezhny). Program Linear differential equations: Riemann­Hilb ert problem, isomono dromic deformations and Painlevґ equations e 1. Riemann­Hilbert problem 2. Vector bundles with connection 3. Isomonodromy deformations 4. Schlesinger equation 5. Painleveґ equations e [2] Diferential equations and isomonodronic deformations (scientific seminar for students in HSE, joint with V.A. Poberezhny) [3] Analitic theory of differential equations (scientific seminar in Steklov Mathematical Institute, joint with V.P. Lexin, R.R. Gontsov, A.V. Klimenko) I have teached the exercises of Calculus, ODE and PDE. I am a supervisor of 5 students. Student Victoria Malyasova has passed your graduate thesis under my supervision.