Number-theoretic method in approximate analysis

S. S. Demidov ; E. A. Morozova ; V. N. Chubarikov ; I. Yu. Rebrova ; I. N. Balaba ; N. N. Dobrovol'skii ; N. M. Dobrovol'skii ; L. P. Dobrovol'skaya ; A. V. Rodionov ; O. A. Pikhtil'kova

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 6-85

Résumé

Into the image it is considered issues of history and the modern development of number-theoretic method in the approximate analysis which based in the work of N. M. Korobov and his disciples. It is reviewed the connection of the theory of uniform distribution and theoretical-numeric method in approximate analysis. It is shown that the condition for the theoretical-numeric method was the integral criterion G. Weyl. It is disassembled main types of number-theoretic nets: uneven, parallelepipedal and algebraic. It is consecrated the activities of the workshop three K, it is explored the biographical information about N. M. Korobov and brief information about the leaders of the seminar and its participants. It is described the main directions of research in theoretical-numeric method in approximate analysis. It is examined the issues of information security theoretic-numeric method in approximate analysis using POIS TMK. More detailed it is outlined the issues of finding the optimal coefficients for parallelepipedal nets, the theory of the hyperbolic Zeta function of lattices, the theory of algebraic nets and its relationship with the theory of Diophantine approximations. In particular, we discuss the algebraic theory of polynomials Tue. The theory is based on the study of submodules of $\mathbb Z[t]$-module $\mathbb Z[t]^2$. It is considered of submodules that are defined by one defining relation and one defining relation $k$-th order. More complex submodule is the submodule given by one polynomial relation. Sub par Tue $j$-the order are directly connected with polynomials Tue $j$-th order. Using the algebraic theory of pairs of submodules of Tue $j$-th order is managed to obtain a new proof of the theorem of M. N. Dobrowolski (senior) that for each $j$ there are two fundamental polynomial Tue $j$-th order, which are expressed through others. Basic polynomials are determined with an accuracy of unimodular polynomial matrices over the ring of integer polynomials. It is discussed the fractional-linear transformation of TDP-forms. It is shown that the transition from TDP-forms associated with an algebraic number $\alpha$ to TDP-the form associated with the residual fraction to algebraic number $\alpha$, TDP-form is converted under the law, similar to the transformation of minimal polynomials and the numerators and denominators of the respective pairs of Tue is converted using the linear-fractional transformations of the second kind. Besides, we discuss the new classification of purely real algebraic irrationalities which based on their expansion in continued fractions. It is shown that for purely real algebraic irrationalities $\alpha$ of degree $n\ge2$, starting from some numbers $m_0=m_0(\alpha)$, the sequence of residual fractions $\alpha_m$ is a sequence given the algebraic irrationalities. It is found recurrence the formula for finding the minimal polynomials of the residual fractions using the linear-fractional transformations. The compositions of these linear-fractional transformations is a linear-fractional transformation that maps the system conjugate to algebraic irrationascenic spots $\alpha$ in the system of associated to the residual fraction, with a pronounced effect of concentration nearly rational fraction $-\frac{Q_{m-2}}{Q_{m-1}}$. It is established that the sequence of minimal polynomials for residual fractions forms a sequence of polynomials with equal discriminants. Lists some of the most pressing unsolved problems.

Détail

Keywords: number-theoretic method, uniform distribution,nonuniform grid, parallelepipedal nets, algebraic nets, hyperbolic dzeta-function of lattices, algebraic theory of polynomials Tue, given an algebraic irrationality, the classification of purely real algebraic irrationalities.