Related papers: Quasi-random multilinear polynomials

We suggest an upper bound on binomial coefficients that holds over the entire parameter range and whose form repeats the form of the de Moivre-Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the…

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…

We give a bound for the number of real solutions to systems of n polynomials in n variables, where the monomials appearing in different polynomials are distinct. This bound is smaller than the fewnomial bound if this structure of the…

In this paper, we study inequalities involving polynomials and quasimodular forms. More precisely, we focus on the monotonicity of the functions of the form $t \mapsto t^m F(it)$ where $F$ is a quasimodular form and $m > 0$. As an…

Let H(N) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in H(N) whose largest coefficients are as small as possible and also for…

We consider random polynomials of the form $G_n(z):= \sum_{|\alpha|\leq n} \xi^{(n)}_{\alpha}p_{n,\alpha}(z)$ where $\{\xi^{(n)}_{\alpha}\}_{|\alpha|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,\alpha}\}_{|\alpha|\leq n}$ form…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m} a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m} |a_{\alpha}|^{r}…

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as…

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…

We show that, for a system of univariate polynomials given in sparse encoding, we can compute a single polynomial defining the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine…

We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $\Pi^*_n$ be the subset of polynomials of degree at most $n$ in $d$…

In this note we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [8, 5] provides us with an upper bound on the…

We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with…

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…

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…

It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In…

We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde…

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…

Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is H{\"o}lder…