Related papers: A polynomial with a root mod $p$ for every $p$ has…

Given a polynomial $p$ of degree $d$ and a bound $\kappa$ on a condition number of $p$, we present the first root-finding algorithms that return all its real and complex roots with a number of bit operations quasi-linear in $d…

Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in…

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In…

We characterize all monic polynomials $f(x) \in \mathbb{Z}[x]$ that have the property that \[f(p) \mid f(p^{p}),~\text{for all sufficiently large primes }p \geq N(f). \] We also give necessary conditions and a sufficient condition for monic…

We show that the roots of any smooth curve of polynomials with only real roots can be parametrized twice differentiably (but not better).

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known…

Let $p$ be a prime and let $g(p)$ be the least primitive root modulo $p$. We prove that for any $\epsilon>0$ and $p$ large enough the bound \begin{align} g(p)\ll p^{\frac{1}{4\sqrt{e}}+\epsilon} \nonumber \end{align} holds for most prime…

Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…

Suppose that f has continuous derivatives thru order r+1 for x>0, and let P_{c} denote the Taylor polynomial to f of order r at x=c,c>0. In a previous paper of the author, it was shown that if r is an odd whole number and the (r+1)st…

The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…

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 attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational…

We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective…

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a…

The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible…

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coeffcients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these…

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…

We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic…

We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes,…