arXiv subjects

Alina Firicel

Publications and source records attributed to Alina Firicel.

3 recordsLinked to original sources

arXiv2011

Rational approximations to algebraic Laurent series with coefficients in a finite field

In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.

math.NT↗

arXiv2010

Sur le développement en fraction continue d'une généralisation de la cubique de Baum et Sweet

In 1976, Baum and Sweet gave the first example of a power series that is algebraic over the field $\mathbb F_2(T)$ and whose continued fraction expansion has partial quotients with bounded degree. This power series is the unique solution of the equation $TX^3+X-T=0$. In 1986, Mills and Robbins described an algorithm that allows to compute the continued fraction expansion of the Baum--Sweet power series. In this paper, we consider the more general equations $TX^{r+1}+X-T=0$, where $r$ is a power of a prime number $p$. Such an equation has a unique solution in the field $\mathbb F_p((T^{-1}))$. Applying an approach already used by Lasjaunias, we give a description of the continued fraction expansion of these algebraic power series.

math.NT↗

arXiv2010

Subword complexity and Laurent series with coefficients in a finite field

Decimal expansions of classical constants such as $\sqrt2$, $π$ and $ζ(3)$ have long been a source of difficult questions. In the case of Laurent series with coefficients in a finite field, where no carry-over difficulties appear, the situation seems to be simplified and drastically different. On the other hand, Carlitz introduced analogs of real numbers such as $π$, $e$ or $ζ(3)$. Hence, it became reasonable to enquire how "complex" the Laurent representation of these "numbers" is. In this paper we prove that the inverse of Carlitz's analog of $π$, $Π_q$, has in general a linear complexity, except in the case $q=2$, when the complexity is quadratic. In particular, this implies the transcendence of $Π_2$ over $\F_2(T)$. In the second part, we consider the classes of Laurent series of at most polynomial complexity and of zero entropy. We show that these satisfy some nice closure properties.

math.NT↗