Related papers: Sharp Remez inequality

We show that for a real polynomial of degree $n$ in two variables $x$ and $y$, any local "sharp turn" must have its "size" $\gtrsim e^{-Cn^{2}}$. We also show that there is indeed an example that has a sharp turn of size $\lesssim e^{-Cn}$.…

For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials,…

We show that the $\Lp$ Busemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo-Nirenberg…

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…

In this paper, we investigate the solubility of homogeneous polynomial equations. The work of Browning, Le boudec, Sawin [3] shows that almost all homogeneous equations of degree $d\geq 4$ in $d+1$ or more variables satisfy the Hasse…

The Sendov conjecture asserts that if $p(z) = \prod_{j=1}^{N}(z-z_j)$ is a polynomial with zeros $|z_j| \leq 1$, then each disk $|z-z_j| \leq 1$ contains a zero of $p'$. Our purpose is the following: Given a zero $z_j$ of order $n \geq 2$,…

An explicit subconvex bound for the Riemann zeta function $\zeta(s)$ on the critical line $s=1/2+it$ is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-Landau lemma. After accounting for the needed correction…

This article is devoted to the sharp improvement of the classical Bohr inequality for bounded analytic functions defined on the unit disk. We also prove two other sharp versions of the Bohr inequality by replacing the constant term by the…

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz…

We derive sharp, explicit constants in inverse trace inequalities for polynomial functions belonging to $\mathbb{P}_p(T)$ (polynomial space with total degree $p$) that are orthogonal to the lower-order subspace $\mathbb{P}_n(T)$, $n\leq p$,…

For $1<p<\infty$, we prove that a necessary and sufficient condition for $s$ to be a zero of the Riemann zeta function in the strip $0<\Re s<1$ is that $$\left(\begin{array}{cccccc} 1 & \frac{1}{3} & \frac{1}{5} & \frac{1}{7} & \frac{1}{9}…

This paper studies the complexity of matrix Putinar's Positivstellens{\"a}tz on the semialgebraic set that is given by the polynomial matrix inequality. \rev{When the quadratic module generated by the constrained polynomial matrix is…

We are interested in finding a nonlinear polynomial $P$ on $\mathbb{R}^n$ that solves the minimal surface equation. Even though no explicit solution is found in this article, we investigate constraints that a polynomial solution must obey.…

For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving…

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…

We study the decision version of tensor spectral norm from the viewpoint of real algebraic complexity. For a rationally specified tensor, the tensor spectral threshold problem asks whether its spectral norm exceeds a prescribed rational…

In this paper we study the Remez-type inequalities for trigonometric polynomials with harmonics from hyperbolic crosses. The interrelation between the Remez and Nikolskii inequalities for individual functions and its applications are…

We prove a sharp Rogers-Shephard type inequality for the p-difference body of a convex body in the two-dimensional case, for every p greater than or equal to one.

Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Korian…

This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, $R$. The subsequences $\{\beta^R_{i}(M)\}_{i\geq 0}$ with even, respectively, odd $i$ are known to be eventually given…