2,593 research outputs found
Computing solutions of linear Mahler equations
International audienceMahler equations relate evaluations of the same function at iterated th 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 -units via frey-hellegouarch curves
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
We investigate the continued fraction expansion of the infinite products
where polynomials satisfy
and . We construct relations between partial quotients of
which can be used to get recurrent formulae for them. We provide that
formulae for the cases and . As an application, we prove that for
where is an arbitrary rational number except 0 and 1, and for
any integer with such that the irrationality exponent
of equals two. In the case we provide a partial analogue of the
last result with several collections of polynomials giving the
irrationality exponent of strictly bigger than two.Comment: 25 page
Extended Rate, more GFUN
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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