The relationship between algebraic equations and $(n,m)$-forms, their degrees and recurrent fractions

Abstract

Algebraic and recursion equations are widely used in different areas of mathematics, so various objects and methods of research that are associated with them are very important. In this article we investigate the relationship between $(n,m)$-forms with generalized Diophantine Pell's equation, algebraic equations of $n$ degree and recurrent fractions. The properties of the $(n,m^n+1)$-forms and their characteristic equation are considered. The author applied parafunctions of triangular matrices to the study of algebraic equations and corresponding recurrence equations. The form of adjacent roots of the annihilating polynomial of arbitrary $(n,m)$-forms over the field of rational numbers are explored.

The following question is very important for some applied problems: Is a given form the largest by module among its adjacent roots? If it is so, then there is a periodic recurrence fraction of $n$-order that is equal to this $(n,m)$-form, and its $m$th rational shortening will be its rational approximation. The author has identified the class $(n,m)$-forms with the largest module among their adjacent roots and showed how to find periodic recurrence fractions of $n$-order and rational approximations for them.