Related papers: Simultaneous approximation by conjugate algebraic …

We generalize Gel'fond's transcendence criterion to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the multiplicative group C* of the field of complex numbers.

We define a transcendence degree for division algebras, by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the…

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…

We count the algebraic numbers of fixed degree by their $\mathbf{w}$-weighted $l_p$-norm which generalizes the na\"ive height, the length, the Euclidean and the Bombieri norms. For non-negative integers $k,l$ such that $k+2l\leq n$ and a…

For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of…

Let xi be a real number which is neither rational nor quadratic over Q. Based on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any real number theta, there exist a constant c>0 and infinitely many non-zero…

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…

We establish an effective improvement on the Liouville inequality for approximation to complex non-real algebraic numbers by quadratic complex algebraic numbers.

In this paper, we improve some transcendence results for $p$--adic continued fractions. In particular, we prove that palindromic and quasi--periodic $p$--adic continued fractions converge either to transcendental numbers or quadratic…

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition…

It is well known that the $(-1)$-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct $q$-analogues of the…

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field…

In this paper, I give sufficient conditions for any linear combination in $\mathbb{Q}$ of numbers $\sum_{n=1}^{\infty}\frac{b_{1,n}}{\alpha_{1,n}}$, $\ldots$, $\sum_{n=1}^{\infty}\frac{b_{K,n}}{\alpha_{K,n}}$ to have algebraic degree…

We study the values of the recently introduced involution J (jimm) of the real line, which is equivariant with the action of the group PGL(2,Z). We test our conjecture that this involution sends algebraic numbers of degree at least three to…

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) =…

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…

We enumerate height-restricted path diagrams associated with $q$-tangent and $q$-secant numbers by considering convergents of continued fractions, leading to expressions involving basic hypergeometric functions. Our work generalises some…

Let $\xi$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. Let $(u_n)_{n \geq 1}$ be a non-degenerate linear recurrence sequence of integers, which is not a polynomial sequence. We…

There are abundant results on Diophantine approximation over fields of positive characteristic (see the survey papers [13, 25]), but there is very little information about simultaneous approximation. In this paper, we develop a technique of…