Related papers: Polynomial Functions on a Central Relation

A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings (V,otimes,oplus,0,1) instead of rings or fields makes traditional approaches impossible. Earlier work shows that the…

This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a general field $F$; in…

Let $k\in \mathbb{N}\setminus\{0\}$. For a commutative ring $R$, the ring of dual numbers of $k$ variables over $R$ is the quotient ring $R[x_1,\ldots,x_k]/ I $, where $I$ is the ideal generated by the set $\{x_ix_j\mid i,j=1,\ldots,k\}$.…

We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth order linear differential equations, and one of the families is…

Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $f\colon G\to E$. A function $f\in C(G,E)$ is a generalized polynomial, if there is an $n\ge 0$ such…

Let $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$. Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge 4^{d+2}d^2R^5$. We prove…

In this paper, a sequence of linear combination of $R_{I}$ type polynomials such that the terms in this sequence have a common zero is constructed. A biorthogonality relation arising from such a sequence is discussed. Besides a sequence of…

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive…

Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,...,p_s such that P= \{x \in R^d : p_1(x) \ge 0, >..., p_s(x) \ge 0 \}; in this case p_1,...,p_s are said to represent P. Denote by $n$ the…

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…

In this paper, we first consider the graph of $(F_1,F_{2},\cdots,F_{n})$ on $\overline{\mathbb{D}}^{n},$ where $F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\cdots,n,$ which has non-isolated CR-singularities if $m_{j}>1$ for some…

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for…

We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$.

We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$…

This contribution aims to obtain several connection formulae for the polynomial sequence, which is orthogonal with respect to the discrete Sobolev inner product \[ \langle f, g\rangle_n=\langle {\bf u}, fg\rangle+ \sum_{j=1}^M \mu_{j}…

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…

We present a functorial construction which, starting from a congruence $\alpha$ of finite index in an algebra A, yields a new algebra C with the following properties: the congruence lattice of C is isomorphic to the interval of congruences…

We study the functional equation $A\circ X=X\circ B$, where $A,$ $B$, and $X$ are polynomials over $\mathbb C$. Using previous results of the author about polynomials sharing preimages of compact sets, we show that for given $B$ its…

For a simple Lie algebra g we consider an analogue of the affine algebra ^gk with n singularities, defined starting from the ring of functions on the n-pointed disk. We study the center of its completed enveloping algebra and prove an…

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…