We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux....

We show how an old principle, due to Walsh (1922), can be used in order to construct an algorithm which finds the roots of polynomials with complex coefficients. This algorithm uses a linear command. From the very first step, the zero is located inside a disk, so several zeros can be searched at the same time.

For $\mathbf{r}=(r_0,...,r_{d-1})\in {ℝ}^d$ define the function$${\tau }_{\mathbf{r}}:{\mathbb{Z}}^d\to {\mathbb{Z}}^d,\mathbf{z}=(z_0,...,z_{d-1})\mapsto (z_1,...,z_{d-1},-⌊\mathbf{rz}⌋),$$where $\mathbf{rz}$ is the scalar product of the vectors $\mathbf{r}$ and $\mathbf{z}$. If each orbit of ${\tau }_{\mathbf{r}}$ ends up at $\mathbf{0}$, we call ${\tau }_{\mathbf{r}}$ a shift radix system. It is a well-known fact that each orbit of ${\tau }_{\mathbf{r}}$ ends up periodically if the polynomial $t^d+r_{d-1}{t}^{d-1}+\cdots +r_0$ associated to $\mathbf{r}$ is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of...

We give an automata-theoretic description of the algebraic closure of the rational function field ${𝔽}_q\left(t\right)$ over a finite field ${𝔽}_q$, generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over ${𝔽}_q$. In passing, we obtain a characterization of well-ordered sets of rational numbers whose base $p$ expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive...

The canonization theorem says that for given $m,n$ for some $m^{*}$ (the first one is called $ER(n;m)$) we have for every function $f$ with domain ${\left[1,\cdots ,m^{*}\right]}^n$, for some $A\in {\left[1,\cdots ,m^{*}\right]}^m$, the question of when the equality $f\left(i_1,\cdots ,i_n\right)=f\left(j_1,\cdots ,j_n\right)$ (where $i_1<\cdots <i_n$ and $j_1<\cdots j_n$ are from $A$) holds has the simplest answer: for some $v\subseteq \{1,\cdots ,n\}$ the equality holds iff ${\bigwedge }_{\ell \in v}{i}_{\ell }=j_{\ell }$. We improve the bound on $ER(n,m)$ so that fixing $n$ the number of exponentiation needed to calculate $ER(n,m)$ is best possible.