Related papers: Some new results on negative polynomial Pell's equ…

In this note, it is shown that the differential polynomial of the form $Q(f)^{(k)}-p$ has infinitely many zeros, and particularly $Q(f)^{(k)}$ has infinitely many fixed points for any positive integer $k$, where $f$ is a transcendental…

We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $\zeta$ of $P$ such that…

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…

For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell…

Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,...,p_s such that P= \{x \in R^d : p_1(x) \ge 0, >..., p_s(x) \ge 0 \}; in this case p_1,...,p_s are said to represent P. Denote by $n$ the…

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em…

We study about solutions of certain kind of non-linear differential difference equations $$f^{n}(z)+wf^{n-1}(z)f^{'}(z)+f^{(k)}(z+c)=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ and…

We describe the limit zero distributions of sequences of polynomials with positive coefficients.

We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of…

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery,…

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…

It is important in drawing techniques to find combinations of two straight lines and their angle bisectors whose slopes are all rational numbers. This problem is reduced to solving the Diophantine equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$…

In this paper, we consider central complete and incomplete Bell polynomials which are generalizations of the recently introduced central Bell polynomials and central analogues for the complete and incomplete Bell polynomials. We investigate…

Let $p\equiv 1\bmod 4$ be a prime number. We use a number field variant of Vinogradov's method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\mathrm{Cl}(-4p)$ of the imaginary…

We find a new lower bound for the maximal number of zeros to harmonic polynomials, $p(z)+\overline{q(z)}$, when ${\rm deg}\, p = n$ and ${\rm deg}\, q = n-2$.

Using non-archimedean q-integrals on Zp defined in [15, 16], we define a new Changhee q-Euler polynomials and numbers which are different from those of Kim [7] and Carlitz [2]. We define generating functions of multiple q-Euler numbers and…

In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.

A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, ..., x_{n - k + 1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily…

Let $\mathbb{Z}_p[x]$ be the set of all functions whose coefficients are in the field of $p$-adic integers $\mathbb{Z}_p$. This work considers a problem of finding a root of a polynomial equation $P(x)=0$ where $P(x)\in\mathbb{Z}_p[x]$. The…

In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $ax^2-by^2=c$.