相关论文: On characterization of Monogenic number fields ass…

For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their…

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$,…

We show that if $A$ is a finite set of non-negative integers then the number of zeros of the function \[ f_A(\theta) = \sum_{a \in A} \cos(a\theta), \] in $[0,2\pi]$, is at least $(\log \log \log |A|)^{1/2-\varepsilon}$. This gives the…

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities…

Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related…

Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient…

In this thesis we study when a homogeneous polynomial $f$ decomposes or "splits" additively. Up to base change this means that it is possible to write $f = g + h$ where $g$ and $h$ are polynomials in independent sets of variables. This…

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold…

Let $F(X)= \prod_{i=1}^k(a_iX+b_i)$ be a polynomial with $a_i, b_i$ being integers. Suppose the discriminant of $F$ is non-zero and $F$ is admissible. Given any natural number $N$, let $S(F,N)$ denotes those integers less than or equal to…

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…

Let $f(x) = \sum\limits _{i=0}^{n} a_i x^i $ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0 < a_0 \leq a_{1} \leq \cdots \leq a_{k-1} < a_{k} < a_{k+1} \leq \cdots \leq a_n$ for some $k$,…

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The…

We find the minimal dimension for a truncated polynomial algebra over an arbitrary field for which there exists a "non-thin" subalgebra. Moreover, we discuss examples of subalgebras, and count them in low dimensions.

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding…

Let $K$ be a field and $f _{n}(X) = (X + 1) ^{n} + (-1) ^{n}(X ^{n} + 1) \in K[X]$, for each $n \in \mathbb N$. This note shows that the polynomials $f _{m}(X)$ and $f _{m'}(X)$ are relatively prime, for some distinct indices $m$ and $m…

Let $A$ be an integral domain with quotient field $K$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra. Denote by $D(F)$ the discriminant of a polynomial $F\in A[X]$. Further, given a finite etale algebra $\Omega$,…

We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so,…

For positive integers $a$ and $b$, we let $[U_n]$ be the Lucas sequence of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{ for $n\ge 2$},\] and let $\pi(m):=\pi_{(a,b)}(m)$ be the…