相关论文: On the maxima of Littlewood polynomials on $[-1,1]…

Littlewood asked how small the ratio $||f||_4/||f||_2$ (where $||.||_\alpha$ denotes the $L^\alpha$ norm on the unit circle) can be for polynomials $f$ having all coefficients in $\{1,-1\}$, as the degree tends to infinity. Since 1988, the…

Polynomials with all the coefficients in $\{ 0,1\}$ and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in $\{ -1,1\}$ are called Littlewood polynomials. By exploiting an algorithm developed…

We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…

Polynomials with coefficients in $\{-1,1\}$ are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vall\'ee Poussin sums,…

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…

Let $f = \sum_{k=0}^n \varepsilon_k z^k$ be a random polynomial, where $\varepsilon_0,\ldots ,\varepsilon_n$ are iid standard Gaussian random variables, and let $\zeta_1,\ldots,\zeta_n$ denote the roots of $f$. We show that the point…

For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda…

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is…

A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.…

We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which…

Littlewood investigated polynomials with coefficients in $\{-1,1\}$ (Littlewood polynomials), to see how small their ratio of norms $||f||_4/||f||_2$ on the unit circle can become as $deg(f)\to\infty$. A small limit is equivalent to slow…

We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given…

A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erd\H{o}s, that ask for Littlewood polynomials that provide a…

If f(x_1, x_2, ..., x_n) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version…

Let $f$ be sampled uniformly at random from the set of degree $n$ polynomials whose coefficients lie in $\{ \pm 1\}$. A folklore conjecture, known to hold under GRH, states that the probability that $f$ is irreducible tends to $1$ as $n$…

Let $k\leq n$. Each polynomial $p\in\oR[x_1,...,x_n]$ can be uniquely written as $p=\sum_{\mu}\mu p_{\mu}$, where $\mu$ ranges over the set $M$ of all monomials in $\oR[x_1,...,x_k]$ and where $p_{\mu}\in\oR[x_{k+1},...,x_n]$. If $p$ is…

We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| < 1\}$. For every pair $(k, n) \in…

Consider a degree-$d$ polynomial $f(\xi_1,\dots,\xi_n)$ of independent Rademacher random variables $\xi_1,\dots,\xi_n$. To what extent can $f(\xi_1,\dots,\xi_n)$ concentrate on a single point? This is the so-called polynomial…

Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can grow with n. We consider such polynomials for odd square-free n, where \phi(n)…

Littlewood polynomials are polynomials with each of their coefficients in {-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…