In 1848 Ch. Hermite formulated a question on periodic representation of cubic numbers. In this paper we introduce a new jacobi-perron type algorithm that provides periodicity for all cubic irrationalities in the input...
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In 1848 Ch. Hermite formulated a question on periodic representation of cubic numbers. In this paper we introduce a new jacobi-perron type algorithm that provides periodicity for all cubic irrationalities in the input and prove the periodicity. This is the first subtractive algorithm for which the algebraic periodicity is proven. The algorithm is designed for the totally real case, providing the answer to Hermite's original question in this case.
We study a new connection between multidimensional continued fractions, such as jacobi-perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables o...
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We study a new connection between multidimensional continued fractions, such as jacobi-perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposables to formulate our conclusions. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study multidimensional continued fraction algorithms throughout the field of formal power series. In this case, we establish a relation between the jacobi-perron algorithm and the version of it introduced by Dubois...
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We study multidimensional continued fraction algorithms throughout the field of formal power series. In this case, we establish a relation between the jacobi-perron algorithm and the version of it introduced by Dubois. Regarding the periodicity of the jacobi-perron algorithm, we define periodic vectors whose coordinates belong to certain finite degree extension fields. We prove also that the convergence of Brun algorithm in the case of multidimensional continued fractions over the Field of Formal Power Series is not exponential.
We give a combinatorial interpretation of vector continued fractions obtained by applying the jacobi-perron algorithm to a vector of p >= 1 resolvent functions of a banded Hessenberg operator of order p + 1. The in...
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We give a combinatorial interpretation of vector continued fractions obtained by applying the jacobi-perron algorithm to a vector of p >= 1 resolvent functions of a banded Hessenberg operator of order p + 1. The interpretation consists in the identification of the coefficients in the power series expansion of the resolvent functions as weight polynomials associated with Lukasiewicz lattice paths in the upper half-plane. In the scalar case p = 1 this reduces to the relation established by P. Flajolet and G. Viennot between jacobi-Stieltjes continued fractions, their power series expansion, and Motzkin paths. We consider three classes of lattice paths, namely the Lukasiewicz paths in the upper half-plane, their symmetric images in the lower half-plane, and a third class of unrestricted lattice paths which are allowed to cross the x-axis. We establish a relation between the three families of paths by means of a relation between the associated generating power series. We also discuss the subcollection of Lukasiewicz paths formed by the partial p-Dyck paths, whose weight polynomials are known in the literature as genetic sums or generalized Stieltjes-Rogers polynomials, and express certain moments of bi-diagonal Hessenberg operators.
In this paper, we study the jacobi-perron algorithm of (alpha, alpha(2)) and (1/alpha, 1/alpha(2)) where alpha is the unique real root of monic cubic irreducible polynomials in certain infinite families. We also inves...
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In this paper, we study the jacobi-perron algorithm of (alpha, alpha(2)) and (1/alpha, 1/alpha(2)) where alpha is the unique real root of monic cubic irreducible polynomials in certain infinite families. We also investigate the associated Hasse-Bernstein units, along with when they are fundamental units in Z[alpha] and Q(alpha).
Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers Q(p). Here, we study the use of multidimensional continued fractions (MC...
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Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers Q(p). Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in R by jacobi and perron as a generalization of continued fractions and they have been recently defined also in Q(p). We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime. Moreover, given algebraically dependent padic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the p-adic jacobi-perron algorithm when it processes some kinds of Q-linearly dependent inputs.
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are...
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The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.& COPY;2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by-nc -nd /4 .0/).
Multidimensional continued fractions (MCFs) were introduced by jacobi and perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the ...
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Multidimensional continued fractions (MCFs) were introduced by jacobi and perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the p-adic numbers Q(p). First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of an MCF, and we perform a general study about their convergence in Q(p). In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in Q(p) contrary to the real case where strong convergence is not always guaranteed. Then, we focus on a specific algorithm that, starting from an m-tuple of numbers in Q(p) (p odd), produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized p-adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.
For all integers m >= n >= 2, we exhibit infinite families of purely periodic jacobi-perron algorithm (JPA) expansions of dimension n with period length equal to m along with the associated Hasse-Bernstein units...
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For all integers m >= n >= 2, we exhibit infinite families of purely periodic jacobi-perron algorithm (JPA) expansions of dimension n with period length equal to m along with the associated Hasse-Bernstein units. Some observations on the units of Levesque-Rhin as well as the periodicity of the JPA expansion of (m(1/3), m(2/3)) are also made. (C) 2016 Published by Elsevier Inc.
We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker po...
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We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of jacobi-perron type algorithms.
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