Related papers: Remarks on polynomial parametrization of sets of i…

Let n and d be positive integers, let k be a field and let P(n,d;k) be the space of the polynomials in n variables of degree at most d with coefficients in k. Let B(n,d) be the set of the Bernstein-Sato polynomials of all polynomials in…

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…

An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial…

Let K be a ring and let A be a subset of K. We say that a map f:A \to K is arithmetic if it satisfies the following conditions: if 1 \in A then f(1)=1, if a,b \in A and a+b \in A then f(a+b)=f(a)+f(b), if a,b \in A and a \cdot b \in A then…

Let $K$ be a global field and $n > 1$ an integer. We show $n$ is composite if and only if there is an irreducible polynomial $f(x) \in K[x]$ of degree $n$ which is reducible $q$-adically for all the primes $q$ of $K$.

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…

Let $R$ be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over $R$ is surjective if and only if it is surjective over $\hat{R_{\mathfrak{m}}}$, the completion of $R$ with respect to…

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid…

A Pythagorean n-tuple is an integer solution of x_1^2+...+x_{n-1}^2=x_n^2. For n=4 and n=6, the Pythagorean n-tuples admit a parametrization by a single n-tuple of polynomials with integer coefficients (which is impossible for n=3). For…

The multi-variable Schmidt polynomials are defined by $$ S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k. $$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm 1$, all the coefficients…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…

Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body $K$ is a polytope if there are sufficiently many tilings which contain a tile similar to $K$. Furthermore, we give an example that this can not be…

Let X be an analytic set defined by polynomials whose coefficients a_1,...,a_s are holomorphic functions. We formulate conditions such that for all sequences {a_(1,n)},...,{a_(s,n)} of holomorphic functions converging locally uniformly to…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

We pursue the question how integers can be ordered or partitioned according to their divisibility properties. Based on pseudometrics on $\mathbb{Z}$, we investigate induced preorders, associated equivalence relations, and quotient sets. The…

If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational…

In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.

A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter. The problem: given a propositional logic program P and a non-negative integer k, decide whether P has a stable…