Related papers: Lineability of summing sets of homogeneous polynom…

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

Suppose $P^n_m$ is the blow up of $\mathbb{P}^n$ at a linear subspace of dimension $m$, $\mathcal{L}=\{L_1,\ldots,L_r\}$ is a (not necessarily full) strong exceptional collection of line bundles on $P^n_m$. Let $Q$ be the quiver associated…

This paper discusses spectral synthesis for those modulation spaces $M^{p,q}_s({\mathbf R}^n)$ which form Banach algebras under pointwise multiplication. An important argument will be the ``ideal theory for Segal algebras'' by H. Reiter…

In this paper, we study sequences of positive numbers preserving summability. In particular, the open set property for such a family of sequences is shown. Several classes of sequences preserving summability, including polynomials, sums of…

Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $\P^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no…

This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…

We prove that the super-linearizability of polynomial systems is preserved by all currently known classes of polynomial automorphisms of $\R^n$. We then establish connections between such automorphisms and a sufficient condition for…

Let $p$ be a prime, and $N$ be a positive integer not divisible by $p$. Denote by ${\rm ord}_N(p)$ the multiplicative order of $p$ modulo $N$. Let $\mathbb{F}_q$ represent the finite field of order $q=p^{{\rm ord}_N(p)}$. For $a,…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

We consider the equation $P(Q(x_1,\ldots,x_\nu))=Q(P(x_1),\ldots,P(x_\nu))$ in polynomials over the field of complex numbers and prove that if ${\rm deg}(P)>1$, then it is only solvable in polynomials that are affinely conjugate to…

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach…

We show that multiplication from $L_p\times L_q$ to $L_1$ (for $p,q\in [1,\infty]$, $1/p+1/q=1$) is a uniformly open mapping. We also prove the uniform openness of the multiplication from $\ell_1\times c_0$ to $\ell_1$. This strengthens the…

We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for $\ell \in \mathbb{N}^*$, $s_i\in (0, 1)$ and $p_i \in [1, +\infty)$, $u :…

Let $\mathcal{P}_{K} (^{n}E; F)$ (resp. $\mathcal{P}_{w} (^{n}E; F)$) the subspace of all $P\in \mathcal{P}(^{n}E; F)$ which are compact (resp. weakly continuous on bounded sets). We show that if $\mathcal{P}_{K} (^{n}E; F)$ contains an…

Let the symmetric functions be defined for the pair of integers $\left( n,r\right) $, $n\geq r\geq 1$, by $p_{n}^{\left( r\right) }=\sum m_{\lambda }$ where $m_{\lambda }$ are the monomial symmetric functions, the sum being over the…

We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in…

Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.

We show that given a positive integer $m$, a real number $p\in\left[ 2,\infty\right)$ and $1\leq s<p^{\ast}$ the set of non--multiple $\left( r;s\right)$--summing $m$--linear forms on $\ell_{p}\times\cdots\times \ell_{p}$ contains, except…

It is known that homogeneous polynomials on Banach spaces cannot be hypercyclic, but there are examples of hypercyclic homogeneous polynomials on some non-normable Fr\'echet spaces. We show the existence of hypercyclic polynomials on…