Related papers: The Bertini irreducibility theorem for higher codi…

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials…

We give a proof of Gabber's presentation lemma for finite fields. We use ideas from Poonen's proof of Bertini's theorem to prove this lemma in the special case of open subsets of the affine plane. We then reduce the case of general smooth…

One of our main goals in this paper is to understand the behavior of limit sets of a diverging sequence of Schottky groups in the group of isometries of the N-dimensional hyperbolic space. This leads us to a generalization of a classical…

We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs…

An old conjecture of Erd\H{o}s and R\'enyi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g(x) \in \mathbb{C}[x]$ when its square $g(x)^2$ has a given number of terms. Further conjectures and results arose,…

We show that in the presence of a geometric condition such as non-negative Ricci curvature, the distributional category of a manifold may be used to bound invariants, such as the first Betti number and macroscopic dimension, from above.…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the…

For a prime knot $K$, we give sufficient conditions for the existence of a component $\mathcal{C}$ of the irreducible ${\rm SL}(2,\mathbb{C})$-character variety of $K$ with $\dim\mathcal{C}>1$, and give a lower bound for $\dim\mathcal{C}$.…

In an earlier paper we showed that we can improve results by Emmy Noether and Alexander Ostrowski concerning the reducibility modulo p of absolutely irreducible polynomials with integer coefficients by giving the problem a geometric turn…

We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries,…

We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of…

We present a new version of the Grobman-Hartman's linearization theorem for random dynamics. Our result holds for infinite dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the…

We present a comprehensive analysis of Bredon's trick, a powerful local-to-global extension principle with broad applications across differential geometry and computational topology. Our main contributions include: (1) novel applications to…

Composition principles for reduced moduli are extended to the case of domains in the $n$-dimensional Euclidean space, $n>2$. As a consequence analogues of extremal decomposition theorems of Kufarev, Dubinin and Kirillova in the planer case…

We prove several new Bertini theorems over arbitrary fields and discrete valuation rings.

Possible irreducible holonomy algebras $\g\subset\osp(p,q|2m)$ of Riemannian supermanifolds under the assumption that $\g$ is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are…

We generalize an equidistribution theorem \`a la Bader-Muchnik for operator-valued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries…

We prove by means of advanced pseudo-monotonicity methods an abstract existence result for parabolic partial differential equations with $\log$-H\"older continuous variable exponent nonlinearity governed by the symmetric part of a gradient…

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra.…