Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.     

Higher Hasse–Witt matrices

We prove a number of $p$-adic congruences for the coefficients of powers of a multivariate polynomial $f\left(x\right)$ with coefficients in a ring $R$ of characteristic zero. If the Hasse–Witt operation is invertible, our congruences yield $p$-adic limit formulae which conjecturally describe the Gauss–Manin connection and the Frobenius operator on the unit-root crystal attached to $f\left(x\right)$. As a second application, we associate with $f\left(x\right)$ formal group laws over $R$. Under certain assumptions these formal group laws are coordinalizations of the Artin–Mazur functors.     

Recurrence in the dynamical system (X,〈Ts〉s∈S) and ideals of βS

A dynamical system is a pair $(X,{〈T_s〉}_{s\in S})$, where $X$ is a compact Hausdorff space, $S$ is a semigroup, for each $s\in S$, $T_s$ is a continuous function from $X$ to $X$, and for all $s,t\in S$, $T_s°T_t=T_{st}$. Given a point $p\in \beta S$, the Stone–?ech compactification of the discrete space $S$, $T_p:X\to X$ is defined by, for $x\in X$, $T_p\left(x\right)=p?{lim}_{s\in S}{T}_s\left(x\right)$. We let $\beta S$ have the operation extending the operation of $S$ such that $\beta S$ is a right topological semigroup and multiplication on the left by any point of $S$ is continuous. Given $p,q\in \beta S$, $T_p°T_q=T_{pq}$, but $T_p$ is usually not continuous. Given a dynamical system $(X,{〈T_s〉}_{s\in S})$, and a point $x\in X$, we let $U\left(x\right)=\{p\in \beta S:T_p\left(x\right)\text{ be uniformly recurrent}\}$. We show that each $U\left(x\right)$ is a left ideal of $\beta S$ and for any semigroup we can get a dynamical system with respect to which $K\left(\beta S\right)={?}_{x\in X}U\left(x\right)$ and $c?K\left(\beta S\right)=?\{U\left(x\right):x\in X\text{and}U\left(x\right)\text{is closed}\}$. And we show that weak cancellation assumptions guarantee that each such $U\left(x\right)$ properly contains $K\left(\beta S\right)$ and has $U\left(x\right)?c?K\left(\beta S\right)\ne ?$.