相关论文: A refined realization theorem in the context of th…

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts…

Let $H_n(t)$ denote the classical Rogers-Szeg\"o polynomial, and let $\tH_n(t_1, \ldots, t_l)$ denote the homogeneous Rogers-Szeg\"o polynomial in $l$ variables, with indeterminate $q$. There is a classical product formula for…

Let p be a trigonometric polynomial, nonnegative on the unit circle $\mathbb{T}$. We say that a measure $\sigma$ on $\mathbb{T}$ belongs to the polynomial Szego class, if $d\sigma=sigma'_{ac}d\theta+d\sigma_s$, $\sigma_s$ is singular, and…

We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a…

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function $g$ supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term…

We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…

Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $\alpha\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability…

Given a graph $G$ of order $n$, the $\sigma$-$polynomial$ of $G$ is the generating function $\sigma(G,x) = \sum a_{i}x^{i}$ where $a_{i}$ is the number of partitions of the vertex set of $G$ into $i$ nonempty independent sets. Such…

A special case of the Menshov--Rademacher theorem implies for almost all polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2\pi i \left(x_1…

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular…

For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $\chi_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to…

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established: Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring of the set of subsets of…

Using the projective oscillator representation of sl(n+1) and Shen's mixed product for Witt algebras, Zhao and the second author (2011) constructed a new functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and use the…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most…

We determine all permutation polynomials over F_{q^2} of the form X^r A(X^{q-1}) where, for some Q which is a power of the characteristic of F_q, the integer r is congruent to Q+1 (mod q+1) and all terms of A(X) have degrees in {0, 1, Q,…

We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fej\'ez spectral factorization theorem that any trigonometric univariate…

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set…