On fractional linear transformations of forms A.~Thue--M.\,N.~Dobrovolsky--V.\,D.~Podsypanina

N. M. Dobrovol'skii ; I. N. Balaba ; I. Yu. Rebrova ; N. N. Dobrovol'skii ; E. A. Matveeva

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 54-97

Résumé

The work builds on the algebraic theory of polynomials Tue. The theory is based on the study of submodules of $\mathbb Z[t]$-module $\mathbb Z[t]^2$. Considers 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 $j$-th order are directly connected with polynomials Tue $j$-th order. Using the algebraic theory of pairs of submodules of Tue $j$-th order 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. In the work introduced linear-fractional conversion 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. Bibliography: 37 titles.

Détail

Keywords: the minimum polynomial of the given algebraic irrationality, residual fractions, continued fractions, TDP-shape, the modules Tue, couple Tue, linear-fractional transformation of the second kind.