Related papers: Simultaneous Polynomial Recurrence

Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and…

Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where…

Let $\gamma(t)=(P_1(t),\ldots,P_n(t))$ where $P_i$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+\gamma(t)\}$ in sets of positive density $\epsilon$ in…

In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops $A\subseteq B$ and a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a polytope…

If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for some prime…

We consider the higher order Tur\'an inequality and higher order log-concavity for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}}…

In this paper, we deal with the question; under what conditions the points $P_i(xi,yi)$ $(i = 1,\cdots, n)$ form a convex polygon provided $x_1 < \cdots < x_n$ holds. One of the main findings of the paper can be stated as follows: "Let…

Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the…

This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…

For $d \geq 2$ and $n \in \mathbb{N}$ even, let $p_n = p_n(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^d$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n \in…

We prove a probabilistic Fourier extension theorem that says Fourier extension holds when averaged over certain smooth Alpert multipliers. The proofs use smooth Alpert wavelets with the classical techniques of stationary phase and…

We prove that any extended formulation that approximates the matching polytope on $n$-vertex graphs up to a factor of $(1+\varepsilon)$ for any $\frac2n \le \varepsilon \le 1$ must have at least $\binom{n}{{\alpha}/{\varepsilon}}$ defining…

We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the…

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the…

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with…

Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it…

Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf…

In this thesis, we answer several questions about the behaviour of prover-verifier interactions under parallel repetition when quantum information is allowed, and the verifier acts independently in them. We first consider the case in which…

Let $N\ge 4$. We show that, if $x_1,\dots,x_N$ and $y_1,\dots,y_N$ are $N$-tuples of strictly positive numbers whose arithmetic, geometric and harmonic means agree, then \[ \max_j x_j <(N-2)\max_j y_j \quad\text{and}\quad \min_j x_j…

Let $p$ be a prime number. Every two-variable polynomial $f(x_1, x_2)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to $\mathbb Z_p$. Our goal of this paper is…