One considers representation of a polynomial in several variables as the sum of values of univariate polynomials taken at linear combinations of the variables.

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

We show that the discriminant of the generalized Laguerre polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ is a non-zero square for some integer pair $(n,\alpha )$, with $n\ge 1$, if and only if $(n,\alpha )$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is the alternating group $A_n$. For example, we establish that for all but finitely many positive integers $n\equiv 2(mod4)$, the only $\alpha$ for which the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is $A_n$ is $\alpha =n$.

If conditional independence constraints define a family of positive distributions that is log-convex then this family turns out to be a Markov model over an undirected graph. This is proved for the distributions on products of finite sets and for the regular Gaussian ones. As a consequence, the assertion known as Brook factorization theorem, Hammersley–Clifford theorem or Gibbs–Markov equivalence is obtained.

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit...

A set $𝒮=\{x_1,...,x_n\}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_i,x_j)\in 𝒮$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k\left(t\right)$ depending only on $t$, such that if $n\le k\left(t\right)$, then the power LCM matrix $\left({[x_i,x_j]}^t\right)$ defined on any gcd-closed set $𝒮=\{x_1,...,x_n\}$ is nonsingular, but for $n\ge k\left(t\right)+1$, there exists a gcd-closed set $𝒮=\{x_1,...,x_n\}$ such that the power LCM matrix $\left({[x_i,x_j]}^t\right)$ on $𝒮$ is singular. In 1996, Hong proved $k\left(1\right)=7$ and noted $k\left(t\right)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate...

We study integral similitude 3 × 3-matrices and those positive integers which occur as products of their row elements, when matrices are symmetric with the same numbers in each row. It turns out that integers for which nontrivial matrices of this type exist define elliptic curves of nonzero rank and are closely related to generalized cubic Fermat equations.