Related papers: Noncommutative Plurisubharmonic Polynomials Part I…

Let $p(x_1,...,x_n) = p(X), X \in R^{n}$ be a homogeneous polynomial of degree $n$ in $n$ real variables, $e = (1,1,..,1) \in R^n$ be a vector of all ones . Such polynomial $p$ is called $e$-hyperbolic if for all real vectors $X \in R^{n}$…

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the…

We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo…

We define an abelian group homomorphism $\mathscr{F}$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $\mathscr{F}$ in the Schur basis are the…

Let $R$ be a finite commutative ring with $1\ne 0$. The set $\mathcal{F}(R)$ of polynomial functions on $R$ is a finite commutative ring with pointwise operations. Its group of units $\mathcal{F}(R)^\times$ is just the set of all…

Given a system of n homogeneous polynomials in n variables which is equivariant with respect to the canonical actions of the symmetric group of n symbols on the variables and on the polynomials, it is proved that its resultant can be…

We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The…

We consider polynomials of the form $\operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated…

We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in $\mathbb R^n$, and having only finitely many zeros in this set. Following C. Bivi\`{a}-Ausina, we…

Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad…

This article gives a class of Nullstellens\"atze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main…

It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a…

We introduce a new problem on the elementary symmetric polynomials $\sigma_k$, stemming from the constraint equations of some modified gravity theory. For which coefficients is a linear combination of $\sigma_k$ $1/p$-concave, with $0 \leq…

For a polynomial $f = x_1^n + \dots + x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g \in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x})…

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m} a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m} |a_{\alpha}|^{r}…

The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin…

We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The…

Commutative Hilbertian Frobenius algebras are those commutative semi-group objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this…