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If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…

Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with…

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…

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…

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a…

Fink, M\'esz\'aros and St.Dizier showed that the Schubert polynomial $\mathfrak{S}_w(x)$ is zero-one if and only if $w$ avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial $\mathfrak{G}_w(x)$ is…

It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to…

We prove that every polynomial map $(f,g):\mathbb{R}^2\to\mathbb{R}^2$ with nowhere vanishing Jacobian such that $\mathrm{deg}\, f\leq 5$, $\mathrm{deg}\,g \leq 6$ is injective.

For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an…

For any real polynomial $p(x)$ of even degree $n$, Shapiro [{\it Arnold Math. J.} 1(1) (2015), 91--99] conjectured that the sum of the number of real zeros of $(n-1)(p')^2 - np p''$ and the number of real zeros of $p$ is positive. We…

Let $\alpha_1,\alpha_2$ be non-zero algebraic numbers such that $\frac{\log \alpha_2}{\log\alpha_1}\notin\mathbb{Q}$ and let $\beta$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime…

Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. In this paper, we find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

We prove that polynomials of degree 10 over finite fields of even characteristic with some conditions on theirs coefficients have a differential uniformity greater than or equal to 6 over $\mathbb{F}_{2^n}$ for all $n$ sufficiently large.

Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$. We establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not…

We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally.

Suppose E/F is a field extension. We ask whether or not there exists an element of E whose characteristic polynomial has one or more zero coefficients in specified positions. We show that the answer is frequently ``no''. We also prove…

This paper considers permutation polynomials over the finite field $F_{q^2}$ in even characteristic by utilizing low-degree permutation rational functions over $F_q$. As a result, we obtain two classes of permutation binomials and six…

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect…