On fractional linear transformations of forms A. Thue–M. N. Dobrovolsky–V. D. Podsypanina

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.