Related papers: Sparse binary cyclotomic polynomials

We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$.…

The maximum gap $g(f)$ of a polynomial $f$ is the maximum of the differences (gaps) between two consecutive exponents that appear in $f$. Let $\Phi_{n}$ and $\Psi_{n}$ denote the $n$-th cyclotomic and $n$-th inverse cyclotomic polynomial,…

In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over…

We show that the equation in the title (with $\Phi_m$ the $m$th cyclotomic polynomial) has no integer solution with $n\ge 1$ in the cases $(m,p)=(15,41), (15,5581),(10,271)$. These equations arise in a recent group theoretical investigation…

We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset\mathbb{R}$, $x_0\in\mathbb{R}\setminus\mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on…

In this paper, we investigate an upper bound of the polar derivative of a polynomial of degree $n$ $$p(z)=(z-z_m)^{t_m} (z-z_{m-1})^{t_{m-1}}\cdots (z-z_0)^{t_0}(a_0+\sum\limits_{\nu=\mu} ^{n-(t_m+\cdots+t_0)} a_{\nu}z^\nu)$$ where zeros…

We show that there exist 0/1 polytopes in R^n with as many as (cn / (log n)^2)^(n/2) facets (or more), where c>0 is an absolute constant.

We prove that for every $\varepsilon>0$ and a nonnegative integer $\omega$ there exist primes $p_1,p_2,\ldots,p_\omega$ such that for $n=p_1p_2\ldots p_\omega$ the height of the cyclotomic polynomial $\Phi_n$ is at least…

In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.

Let $p$ be a prime number. As a standard application of the irreducibility criterion of Eisenstein, it is well known that the $p$-th cyclotomic polynomial $\Phi_p(t)=1+t+\dots+t^{p-1}$ is the minimal polynomial of $e^{2\pi i/p}$ over…

We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the…

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether…

We provide an alternative exposition of a result due to Schinzel. Fix an integer $k \ge 2$. For almost all choices of positive integers $n_{1} < \cdots < n_{k}$, we show that the polynomial $F(x) = 1 + x^{n_{1}} + \cdots + x^{n_{k}}$,…

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…

Let $A_n$ denote the height of cyclotomic polynomial $\Phi_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le \epsilon_k\phi(n)^{k^{-1}2^{k-1}-1}$ with $-\log\epsilon_k\sim c2^k$, $c>0$. The same statement is true…

We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is…

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse cyclotomic polynomial.…

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…