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Abstract
We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction ???????? are concentrated about fractions − ????????−2 ????????−1 is either in the interval of radius ???? (︁ 1 ????2 ????−1 )︁ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number ????0 = ????0(????), fair recurrent formula for incomplete private ???????? expansions of real algebraic irrationals ????, Express ???????? using the values of the minimal polynomial ????????−1(????) for residual fractions ????????−1 and its derivative at the point ????????−1. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of ???? in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − ????????−2 ????????−1 . It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number ???? and its limit points.
Highlights
Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction αm are concentrated about fractions
It is established that, starting from some number m0 = m0(α), fair recurrent formula for incomplete private qm expansions of real algebraic irrationals α, Express qm using the values of the minimal polynomial fm−1(x) for residual fractions αm−1 and its derivative at the point qm−1
Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of α in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction
Summary
История теории цепных дробей насчитывает уже более трехсот лет. В значительной степени основы этой теории были заложены в трудах Л. О разложение алгебраических иррациональностей степени n > 2 в цепные дроби известно очень мало. В работе [10] исследовались минимальные многочлены остаточных дробей разложения вещественных алгебраических иррациональностей в цепные дроби. В этих вопросах существенную роль сыграли дробно-линейные преобразования минимальных многочленов вещественных алгебраических иррациональностей. Целью данной работы — дать новую классификацию чисто-вещественных алгебраических иррациональностей на основе их разложения в цепные дроби и установить предельные соотношения для коэффициентов минимальных многочленов остаточных дробей. Что случай приведённых алгебраических иррациональностей n-ой степени имеет тесную связь с квадратурными формулами с весами в методе К. 108–109) дается описание поведения остаточных дробей и их сопряжённых чисел для разложения алгебраических чисел в цепные дроби. 111–115) строится новая классификация чисто-вещественных алгебраических иррациональностей с точки зрения разложения их в цепные дроби. Что все пропущенные доказательства теорем и лемм содержатся в работах [10], [11]
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