Polynomial Inequalities and Pluripotential Theory

Thursday 31/08/2017 h 9:30-12:30 Room 2BC30 Torre Archimede

ARGOMENTI: Seminari Display

- "How to calculate the complex equilibrium measure?" M. Baran (Department of Applied Mathematics University of Agriculture in Krakow (PL))

Abstract: It is well known that the complex Monge Ampere operator is well defined for locally bounded psh functions and is continuous with respect to monotone sequences of smooth psh functions. We can use this to calculate equiibrium measure for families of compact sets in C^N, among them complex and real balls, and the value of the complex Monge Ampere operator for homogeneous (or logarithmically homogeneous) extremal functions.

- "Markov type inequalities on algebraic hypersurfaces" T. Beberok (")

Abstract: It is well known that if $E$ is a $C^{\infty}$ determining compact set in $\mathbb{R}^n$, then Markov's
inequality for derivatives of polynomials holds on $E$ iff there exists a continuous linear extension operator $L\colon C^{\infty}(E) \rightarrow C^{\infty} (\mathbb{R}^n)$.
The purpose of this talk is to consider Markov type inequalities on compact subset of algebraic hypersurfaces (it's clear that classical Markov inequality does not hold for these sets) in the context of the existence of an extension operator.

- "Admissible meshes on some compact subsets of algebraic hypersurfaces" A. Kowalska


We construct admissible meshes on certain compact subsets of algebraic hypersurfaces of the form $V=\{z_{N+1}^k=s(z_1,\dots ,z_N)\}\subset\mathbb{C}^{N+1}$, where $s$ is a non constant polynomial of $N$ complex variables. The assumptions about these compact subsets is closely related to a Markov inequality for these subsets and structure of the algebraic set V. We will introduce suitable tools on the hypersurfaces V useing some algebraic results.

The talk based on joint papers with M.Baran, T.Beberok, L.Bia?as-Cie? and J.P.Calvi.

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