Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.

In this paper we investigate the factorization of the polynomials $f\left(x\right)x^n+g\left(x\right)\in \mathbb{Z}\left[x\right]$ in the special case where $f\left(x\right)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f\left(x\right)$ is monic and linear.

We prove that the study of the Łojasiewicz exponent at infinity of overdetermined polynomial mappings $ℂⁿ\to {ℂ}^m$, m > n, can be reduced to the one when m = n.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈ (0,1]. Essentially...

Si prova che ogni polinomio in una variabile reale di grado $n$ è somma di $n+1$ funzioni periodiche, ovviamente non tutte continue, e che ci sono funzioni di una variabile reale che non sono somma di un numero finito di funzioni periodiche.

The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences {μk} k=0...∞ which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞. More precisely, it is established that...

For $p$ an odd prime, we show that the Fekete polynomial $f_p\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right)t^a$ has $\sim {\kappa }_{0}p$ zeros on the unit circle, where $0.500813\&gt;{\kappa }_0\&gt;0.500668$. Here ${\kappa }_0-1/2$ is the probability that the function $1/x+1/(1-x)+{\sum }_{n\in \mathbb{Z}:n\ne 0,1}{\delta }_n/(x-n)$ has a zero in $]0,1[$, where each ${\delta }_n$ is $\pm 1$ with y $1/2$. In fact $f_p\left(t\right)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $|f_p\left(e\right(2i\pi (K+\tau )/p\left)\right)|\&lt;ϵ\sqrt{p}$ for some $\tau \in ]0,1[$ then there is a zero of $f$ close to this arc.