Related papers: Sparse systems with high local multiplicity

In this paper, $S$ denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of \emph{\textbf{combinatorial $k$-systoles}} that is closed curves with self-intersection numbers greater than $k$ and…

Let $M$ and $N$ be smooth (real or complex) manifolds, and let $M$ be equipped with some Riemannian metric. A continuous map $f\colon M\longrightarrow N$ admits a local $k$-multiplicity if, for every real number $\omega >0$, there exist $k$…

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria…

Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and…

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central…

This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the…

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most…

Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of…

The notion of cross intersecting set pair system of size $m$, $\Big(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m\Big)$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne\emptyset$, was introduced by Bollob\'as and it became an important tool of extremal…

For any simplicial complex on m vertices a moment-angle complex Z_K embedded in C^m can be defined. There is a canonical action of a torus T^m on Z_K, but this action fails to be free. The Buchstaber number is the maximal integer s(K) for…

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly…

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 study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or…

We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is…

We consider maximum rooted tree extension counts in random graphs, i.e., we consider M_n = \max_v X_v where X_v counts the number of copies of a given tree in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the random…

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…

We study the maximal multiplicity locus of a variety $X$ over a field of characteristic $p>0$ that is provided with a finite surjective radical morphism $\delta:X\rightarrow V$, where $V$ is regular, for example, when…

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $\limsup_{n \to \infty} m/n<1$, with high…

Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $2L$, we develop a local message passing algorithm whose…