Related papers: Algebraic Independence and Mahler's method
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…
We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients…
One studies a particular algebraic system where the unknowns are matrices. We solve this system according to the parameters values thanks to the theory of Grobner basis.
Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already…
The regular independence number, introduced by Albertson and Boutin in 1990, is the size of a largest set of independent vertices with the same degree. Lower bounds were proven for this invariant, in terms of the order, for trees and planar…
We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ for $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ a Sturmian sequence over a binary alphabet and $\beta$ an algebraic number with…
The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
We present a method to compute the Euler characteristic of an algebraic subset of $\bc^n$. This method relies on clasical tools such as Gr\"obner basis and primary decomposition. The existence of this method allows us to define a new…
Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic…
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving…
A. Dubickas and C. Smyth introduced the metric Mahler measure $$ M_1(\alpha) = \inf\left\{\sum_{n=1}^N M(\alpha_n): N \in \mathbb N, \alpha_1 \cdots \alpha_N = \alpha\right\}, $$ where $M(\alpha)$ denotes the usual (logarithmic) Mahler…
In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined…
We present a new completely effective proof of the Lindemann-Weierstrass theorem based on algebraic independence methods. Although it is slightly weaker than the best known estimate due to A. Sert, it improves the best estimate due to M.…
Let $K$ be an algebraically closed field of characteristic 0 and $f\in K[t]$ a polynomial of degree $d\geq 2$. There exists a local conjugacy $\psi_f(t)\in tK[[1/t]]$ such that $\psi_f(t^d)=f(\psi_f(t))$. It has been known that $\psi_f$ is…
For every positive integer $m$ LeVeque (1953) defined the $U_m$-numbers as the transcendental numbers that admit very good approximation by algebraic numbers of degree $m$, but not by those of smaller degree. In these terms, Mahler's…
We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…
The Hoffman ratio bound, Lov\'{a}sz theta function and Schrijver theta function are classical upper bounds for the independence number of graphs, which are useful in graph theory, extremal combinatorics and information theory. By using…
An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some…
We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…