2,593 research outputs found

    Computing solutions of linear Mahler equations

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    International audienceMahler equations relate evaluations of the same function ff at iterated bbth powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently, the problem of solving Mahler equations in closed form has occurred in connection with number-theoretic questions. A difficulty in the manipulation of Mahler equations is the exponential blow-up of degrees when applying a Mahler operator to a polynomial. In this work, we present algorithms for solving linear Mahler equations for series, polynomials, and rational functions, and get polynomial-time complexity under a mild assumption. Incidentally, we develop an algorithm for computing the gcrd of a family of linear Mahler operators

    Sums of two s{s}-units via frey-hellegouarch curves

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    In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables to carry out such a program explicitly, at least for certain small sets of primes S; we do so for S = {2, 3} and S = {3, 5, 7}.Comment: Missing solution in Prop. 5.4 added. To appear in Mathematics of Computatio

    Continued fractions of certain Mahler functions

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    We investigate the continued fraction expansion of the infinite products g(x)=x1t=0P(xdt)g(x) = x^{-1}\prod_{t=0}^\infty P(x^{-d^t}) where polynomials P(x)P(x) satisfy P(0)=1P(0)=1 and deg(P)<d\deg(P)<d. We construct relations between partial quotients of g(x)g(x) which can be used to get recurrent formulae for them. We provide that formulae for the cases d=2d=2 and d=3d=3. As an application, we prove that for P(x)=1+uxP(x) = 1+ux where uu is an arbitrary rational number except 0 and 1, and for any integer bb with b>1|b|>1 such that g(b)0g(b)\neq0 the irrationality exponent of g(b)g(b) equals two. In the case d=3d=3 we provide a partial analogue of the last result with several collections of polynomials P(x)P(x) giving the irrationality exponent of g(b)g(b) strictly bigger than two.Comment: 25 page

    Extended Rate, more GFUN

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    We present a software package that guesses formulae for sequences of, for example, rational numbers or rational functions, given the first few terms. We implement an algorithm due to Bernhard Beckermann and George Labahn, together with some enhancements to render our package efficient. Thus we extend and complement Christian Krattenthaler's program Rate, the parts concerned with guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's qGeneratingFunctions.m.Comment: 26 page
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