Related papers: A Hilbert Lemniscate Theorem in C^2

Motivated by problems of uncertainty propagation and robust estimation we are interested in computing a polynomial sublevel set of fixed degree and minimum volume that contains a given semialgebraic set K. At this level of generality this…

We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a` la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler…

We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $\beta$-density function $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb…

In this paper we study the unitary equivalence between Hilbert modules over a locally C*-algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally C*-algebra and show that a Hilbert module…

We study the density of polynomials in $H^2(E,\varphi)$, the space of square integrable functions with respect to $e^{-\varphi}dm$ and holomorphic on the interior of $E$ in $\mathbb{C}$, where $\varphi$ is a subharmonic function and $dm$ is…

The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring $k[x_0,\dots,x_n]$, in order to design two algorithms: the first one takes as input $n$ and an admissible Hilbert polynomial…

Let $K$ be a complete discretely valued field with residue field $\bar K$ of dimension $1$ (not necessarily perfect). This occurs if and only if $K$ has dimension $2$. We prove the following statements on the arithmetic of such fields: -…

We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n is normal, Cohen-Macaulay, and Gorenstein, and…

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly…

The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety.…

We introduce a new algebraic-cycle model for the motivic cohomology theory of truncated polynomials $k[t]/(t^m)$ in one variable. This approach uses ideas from the deformation theory and non-archimedean analysis, and is distinct from the…

Let $\mathbb C_k[Z_1,\ldots, Z_n]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and $\mathscr{C}_n$ denote the set of all $n$ - tuple $\boldsymbol T=(T_1,\ldots,T_n)$ of commuting contractions on some…

We identify the spaces of homogeneous polynomials in two variables K[Y^k, XY^{k-1}, ..., X^k] among representations of the Lie ring sl(2,K). This amounts to constructing a compatible K-linear structure on some abstract sl(2,K)-modules,…

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$.…

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class $c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with…

This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the $m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb…

The complex Monge-Amp\`ere equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of complex Monge- Amp\`ere equation in the framework…

Let H be a subgroup of some locally compact group G. Assume H is approximable by discrete subgroups and G admits neighborhood bases which are "almost-invariant" under conjugation by finite subsets of H. Let $m: G \to \mathbb{C}$ be a…

We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $$\hat{\mathcal{H}}_N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N \gamma_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k ,$$ with canonical operators…