Related papers: Galois groups and Cantor actions

The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all…

Let $f(x) = ax^d + b \in K[x]$ be a unicritical polynomial with degree $d \geq 2$ which is coprime to $\mathrm{char} K$. We provide an explicit presentation for the profinite iterated monodromy group of $f$, analyze the structure of this…

The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a…

Given a function field $K$ and $\phi \in K[x]$, we study two finiteness questions related to iteration of $\phi$: whether all but finitely many terms of an orbit of $\phi$ must possess a primitive prime divisor, and whether the Galois…

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action…

We study a graded Lie algebra arising from the Galois action on the pro-$p$ fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie…

The Galois group of an infinite Galois extension has a natural topology, called the Krull topology, which has the important property of being profinite. It is impossible to talk about Galois representations, and hence the Langlands Program,…

In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre…

We give a description of the rational representations of the differential Galois group of a Picard-Vessiot extension.

In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study…

In positive characteristic, nearly all Picard-Vessiot extensions are inseparable over some intermediate iterative differential extensions. In the Galois correspondence, these intermediate fields correspond to nonreduced subgroup schemes of…

In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring $k$, via…

Generalized non-autonomous linear celullar automata are systems of linear difference equations with many variables that can be seen as convolution equations in a discrete group. We study those systems from the stand point of the Galois…

Let $G$ be a Lie group acting on a vector space $V$. Given a set of $G$-invariants, one can ask the question : does this set of invariants characterize the group $G$ ? We recall here some known results, ask questions and state some…

Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $\phi(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely…

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the…

In this ongoing work, we extend to a class of well-behaved pre-special hyperfields the work of J. Min\'a\v c and Spira (\cite{minac1996witt}) that describes a (pro-2)-group of a field extension that encodes the quadratic form theory of a…

We look at the asymptotic behavior of the coefficients of the $q$-binomial coefficients (or Gaussian polynomials) $\binom{a+k}{k}_q$, when $k$ is fixed. We give a number of results in this direction, some of which involve Eulerian…

We study a refined version of the Linnik problem on the asymptotic behavior of the number of representations of integer $m$ by an integral polynomial as $m$ tends to infinity. We assume that the polynomial arises from invariant theory, and…

We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.