Related papers: Polynomial analogue of the Smarandache function

Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $\alpha \ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left(…

Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_\lambda(x)$ that we call the partition polynomial for the…

We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the…

Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the…

For an arbitrary set or multiset $A$ of positive integers, we associate the $A$-partition function $p_A(n)$ (that is the number of partitions of $n$ whose parts belong to $A$). We also consider the analogue of the $k$-colored partition…

Consider the following truncated Freud linear functional $\mathbf{u}_z$ depending on a parameter $z$, $$\langle\mathbf{u}_z,p\rangle=\int_0^\infty p(x)e^{-zx^4}dx,\quad z>0.$$ The aim of this work is to analyze the properties of the…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

Let $D$ be a negative integer congruent to $0$ or $1\bmod{4}$ and $\mathcal{O}=\mathcal{O}_D$ be the corresponding order of $ K=\mathbb{Q}(\sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $…

Carlitz rank and index are two important measures for the complexity of a permutation polynomial $f(x)$ over the finite field $\F_q$. In particular, for cryptographic applications we need both, a high Carlitz rank and a high index. In this…

In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial $F$ of degree $d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only if it…

Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n =F_{n-1}+F_{n-2}$ for $n \geq 3$. We let $n_F! = F_1 \cdots F_n$ and $\binom{n}{k}_F = \frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\downarrow_0} = (x)_{\uparrow_0} = 1$ and…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A…

Motivated by a question of Venkataramana, we consider the greatest common divisor of $\phi(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we…

We answer a number of questions relating to the pseudo-Smarandache function Z(n). We show that the ratio of consecutive values $Z(n+1)/Z(n)$ and $Z(n-1)/Z(n)$ are unbounded; that $Z(2n)/Z(n)$ is unbounded; that $n/Z(n)$ takes every integer…

Let $F$ be an affine flat group scheme over a commutative ring $R$, and $S$ an $F$-algebra (an $R$-algebra on which $F$ acts). We define an equivariant analogue $Q_F(S)$ of the total ring of fractions $Q(S)$ of $S$. It is the largest…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with maximal ideal $\frak{m}=(x_1,...,x_n)$, and let $I$ be a graded ideal of $S$. In this paper, we define the saturation number $\sat(I)$ of $I$ to be the…

The aim of this paper is to provide sufficient conditions for when a polynomial or rational function over a field K is prime using its order of vanishing at infinity and the resultant.

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

We consider the class of polynomial optimization problems $\inf \{f(x):x\in K\}$ for which the quadratic module generated by the polynomials that define $K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such problems,…