Related papers: Simultaneous $p$-orderings and minimising volumes …

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which…

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always…

We prove a generalization of the classical Borsuk--Ulam Theorem under small perturbations (shaking) of the sphere. We show that for a generic perturbation of a continuous map $f : S^2 \to \mathbb{R}^2$, the number of points $x \in S^2$ such…

We prove a global existence result with initial data of low regularity, and prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear term but with a possibly large exterior confining potential in…

An interesting result about the existence of "intermediate" set-valued mappings between pairs of such mappings was obtained by Nepomnyashchii. His construction was for a paracompact domain, and he remarked that his result is similar to…

Let $\{p_1,\dots,p_n\}$ and $\{q_1,\dots,q_n\}$ be two sets of $n$ labeled points in general position in the plane. We say that these two point sets have the same order type if for every triple of indices $(i,j,k)$, $p_k$ is above the…

A four-dimensional universe, arising from a flux compactification of Type IIB string theory, contains scalar fields with a potential determined by topological and geometric parameters of the internal -hidden- dimensions. We show that…

We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and M\"uller from the continuous setting to various discrete settings, obtaining sharp bounds for exponential sums and the number of solutions to polynomial congruences for…

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$…

Consider the classical problem of rational simultaneous approximation to a point in $\mathbb{R}^{n}$. The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and…

For $n\geq 1$, we exhibit a lower bound for the volume of a unit vector field on $\mathbb{S}^{2n+1}\backslash\{\pm p\}$ depending on the absolute values of its Poincar\'e indices around $\pm p$. We determine which vector fields achieve this…

In 2001 Wolansky \cite{Wol} introduced a particle number-Casimir functional for the Einstein-Vlasov system. Two open questions are associated with this functional. First, a meaningful variational problem should be formulated and the…

We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the…

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid…

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of…

We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…

The number balancing (NBP) problem is the following: given real numbers $a_1,\ldots,a_n \in [0,1]$, find two disjoint subsets $I_1,I_2 \subseteq [n]$ so that the difference $|\sum_{i \in I_1}a_i - \sum_{i \in I_2}a_i|$ of their sums is…

The classical P\'olya-Tchebotarev problem, commonly stated as a max-min logarithmic energy problem, asks for finding a compact of minimal capacity in the complex plane which connects a prescribed collection of fixed points. Variants of this…

We introduce some generalizations of the Euler-Kronecker constant of a number field and study their arithmetic nature.

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Gr\"obner bases for the ideal of the data points.…