Related papers: On Monogeneity of reciprocal polynomials

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations…

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $n\ge 1$ and $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In…

Let $R$ be a commutative integral domain with unit, $f$ be a nonconstant monic polynomial in $R[t]$, and $I_f \subset R[t]$ be the ideal generated by $f$. In this paper we study the group of $R$-algebra automorphisms of the $R$-algebra…

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

Let K be a field and denote by K[t], the polynomial ring with coefficients in K. Set A = K[f1,. .. , fs], with f1,. .. , fs $\in$ K[t]. We give a procedure to calculate the monoid of degrees of the K algebra M = F1A + $\times$ $\times$…

Let $n \neq 8$ be a positive integer such that $n+1 \neq 2^u$ for any integer $u\geq 2$. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n+1$. Let $a_j(x)$ with…

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic…

Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random…

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…

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…

We study the number of real zeros of trigonometric polynomials in a period and the number of zeros of self-reciprocal algebraic polynomials on the unit circle under the assumption that their coefficients are in a fixed finite set of real…

We identify the dominant part of the Frenkel-Reshetikhin $q$-character with a natural invariant arising from the Langlands/Zelevinsky parameterization for affine Hecke algebras. We introduce the reciprocal character of a module over a…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of…

A polynomial $f(x)$ of degree $d$ is said to be magic positive if all the coefficients are non-negative when $f(x)$ is expanded with respect to the basis $\{x^i(x+1)^{d-i}\}_{i=0}^d$. It is known that if $f(x)$ is magic positive, then the…

Let $K=\mathbb Q(\theta)$ be an algebraic number field with $\theta$ a root of an irreducible trinomial $f(x)=x^6+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, for each prime number $p$ we compute the highest power of $p$ dividing the…

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 prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$…

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…