Related papers: Finiteness Results for Hilbert's Irreducibility Th…

Let $k$ be a field and let $V$ be a $k$-vector space of dimension $d$. Let $G \subseteq GL(V)$ be a finite group. Let $r = \dim_k (V^*)^G$. Assume $r \geq 1$. Let $R = k[V]^G$ be the ring of invariants of $G$. Let $H_R(n) =…

We establish a version "over the ring" of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in $k+n$ variables, with coefficients in $\mathbb Z$, of positive degree in the last $n$ variables, we show that if…

The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this…

We present here quantitative versions in 1 dimension of Faltings'theorem according to which the set of the K-rational points (where K is a given number field) of an abelian variety A definied over K, which are close (with respect to a…

We prove, assuming resolution of singularities in positive characteristic, an analogue of Siegel's theorem on sum of squares in positive characteristic. The method of proof combines techniques from central simple algebras with model theory…

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ…

We establish versions of Szeg\H{o}'s distance formula and Widom's theorem on invertibility of (a family of) Toeplitz operators in a class of finite codimension subalgebras of uniform algebras, obtained by imposing a finite number of linear…

We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a…

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial…

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R…

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear…

This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with…

Let $(K,\mathcal O,k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $\Lambda$ is an $\mathcal O$-order in a separable $K$-algebra, then $\textrm{Pic}_{\mathcal O}(\Lambda)$ carries the structure of an…

Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma$ be a finitely generated Zariski dense subgroup of $G(K)$ and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if…

We study irreducible specializations, in particular when group-preserving specializations may not exist. We obtain a criterion in terms of embedding problems. We include several applications to analogs of Schinzel's hypothesis H and to the…

Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring…

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the…

We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4}…

The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction,…

Green's general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over and infinite field K modulo a general linear form. We strengthen Green's result by showing that the linear…