Abstract. We establish a new transcendence criterion of p-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded p-adic absolute value of partial quotients. This is p-adic analogy of Baker's result. We also prove that p-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.

In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{*}$ for Laurent series over a finite field. We prove that for an integer $n\geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j\geq 1}$ and a sequence of algebraic Laurent series $(\xi_j)_{j\geq 1}$ such that deg $\xi_j=p^{k_j}+1$ and \begin{equation} w_1(\xi_j)=w_1 ^{*}(\xi_j)=\ldots =w_n(\xi_j)=w_n ^{*}(\xi_j)=w \end{equation} for any $j\geq 1$. For each $n\geq 2$, we give explicit examples of Laurent series $\xi $ for which $w_n(\xi )$ and $w_n^{*}(\xi )$ are different.Comment: 22 page

We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.

No abstract