Related papers: The size of semigroup orbits modulo primes

Consider the action of a connected complex reductive group on a finite-dimensional vector space. A fundamental result in invariant theory states that the orbit closure of a vector v is separated from the origin if and only if some…

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a…

Consider a polynomial vector field $\xi$ in $\mathbb{C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of…

Let $F : \mathrm{End}_{\mathbb{F_p}}(\mathbb{G}_{a/K}^d)$ be an additive polynomial mapping over a global function field $K/\mathbb{F}_q$, and let $P \in \mathbb{G}_a^d(K)$. Following Silverman, consider $\delta := \lim_{n \in \mathbb{N}}…

We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining…

Let S be a surface and let Mod(S,K) be the mapping class group of S permuting a Cantor subset K of S. We prove two structure theorems for normal subgroups of Mod(S,K). (Purity:) if S has finite type, every normal subgroup of Mod(S,K) either…

For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(\mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the…

We consider the question of which semigroups can occur as the semigroup $S_R(\nu)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of…

The dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the market point is computed for three families of symmetric semigroups of multiplicity six. We also collect the dimensions of such moduli…

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group…

Let $K$ be a number field, and $\varphi_{1},\ldots,\varphi_{g}\in K(t)$ be finitely many rational maps, each of degree at least $2$. We first show that for generic finite sets $\mathcal{A}_{1},\ldots,\mathcal{A}_{g}$ consisting entirely of…

Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$…

Let ${\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset{\mathbb R}^m$ and ${\mathcal S}^*(M)$ its subring of bounded semialgebraic functions. In this work we compute the size of the fibers…

Let $s\geq 8$ be an integer and $P$ be a set of primes with relative lower density greater than $\sqrt{1-\min\{s,16\}/32}$. We prove that every sufficiently large integer $n\equiv s({\rm mod}24)$ can be represented by a sum of $s$ squares…

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for \'etale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined…

We study the rate of growth of $p(n,S,M)$, the number of partitions of $n$ whose parts all belong to $S$ and whose multiplicities all belong to $M$, where $S$ (resp. $M$) are given infinite sets of positive (resp. nonnegative) integers. We…

In this note, we prove that if $(G,V)$ is a prehomogeneous vector space over any field $k$ such that the stabilizer of a generic point is reductive, the set of semi-stable points is a single orbit over the separable closure of $k$.

In this paper we classify triangular semisimple and cosemisimple Hopf algebras over any algebraically closed field k. Namely, we construct, for each positive integer N, relatively prime to the characteristic of k if it is positive, a…

We investigate the geometry of closed, orientable, hyperbolic $3$-manifolds whose fundamental groups are $k$-free for a given integer $k\ge 3$. We show that any such manifold $M$ contains a point $P$ of $M$ with the following property: If…

We prove that for any distributive join-semilattice S, there are a meet-semilattice P with zero and a map f:PxP-->S such that f(x,z)<=f(x,y)vf(y,z) and x<=y implies that f(x,y)=0, for all x,y,z in P, together with the following conditions:…