We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain K. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta-Siciak extremal function). When the inscribed ellipse method...

Let K be any subset of ${ℂ}^N$. We define a pluricomplex Green’s function $V_{K,\theta }$ for θ-incomplete polynomials. We establish properties of $V_{K,\theta }$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of ${ℝ}^N$, we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K\setminus supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$. We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$ when K is a compact section.

We apply pluripotential theory to establish results in ${ℝ}^k$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $\Sigma \subset {ℝ}^k$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact sections includes...

We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in ${ℝ}^N$ is Hölder continuous then E admits a Szegö type inequality with weight function $dist{(x,\partial E)}^{-(1-\kappa )}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

We define directional Robin constants associated to a compact subset of an algebraic curve. We show that these constants satisfy an upper envelope formula given by polynomials. We use this formula to relate the directional Robin constants of the set to its directional Chebyshev constants. These constants can be used to characterize algebraic curves on which the Siciak-Zaharjuta extremal function is harmonic.

We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function q on an open connected subset X in ℂⁿ. This function is also known as the weighted Green function with logarithmic pole at infinity and weighted global extremal function.

We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in ℂⁿ by Lempert and by Lárusson and Sigurdsson.

Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega$ of dimension $n$. We denote by $H^{\infty }\left(D\right)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_K^D$ a compact subset of the space $C\left(K\right)$ consisted of all restrictions of functions from the unit ball ${𝔹}_{H^{\infty }\left(D\right)}$. In 1950ies Kolmogorov posed a problem: does$${\mathscr{H}}_{\epsilon }\left(A_K^D\right)\sim \tau {\left(ln\frac{1}{\epsilon }\right)}^{n+1},\epsilon \to 0,$$where ${\mathscr{H}}_{\epsilon }\left(A_K^D\right)$ is the $\epsilon$-entropy of the compact $A_K^D$. We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters...

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in ℝⁿ. Another class of extreme singularities is presented by means of a notion of relative type.

Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain generalization...

Let $f$ be a dominant rational map of ${\mathbb{P}}^k$ such that there exists $s\&lt;k$ with ${\lambda }_s\left(f\right)\&gt;{\lambda }_l\left(f\right)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut}\left({\mathbb{P}}^k\right)$, the map $f\circ A$ admits a hyperbolic measure of maximal entropy $log{\lambda }_s\left(f\right)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to ${\mathbb{P}}^{k+1}$. This provides many examples where non uniform hyperbolic dynamics is established.One of the key tools is to approximate the graph of a meromorphic...

We prove that every singular algebraic curve in ℝⁿ admits local tangential Markov inequalities at each of its points. More precisely, we show that the Markov exponent at a point of a real algebraic curve A is less than or equal to twice the multiplicity of the smallest complex algebraic curve containing A.

Les fonctions plurisousharmoniques négatives dans un domaine Ω de ℂⁿ forment un cône convexe. Nous considérons les points extrémaux de ce cône, et donnons trois exemples. En particulier, nous traitons le cas de la fonction de Green pluricomplexe. Nous calculons celle du bidisque, lorsque les pôles se situent sur un axe. Nous montrons que cette fonction ne coïncide pas avec la fonction de Lempert correspondante. Cela donne un contre-exemple à une conjecture de Dan Coman.

A compact set $K\subset {ℂ}^N$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that $V_K\left(z\right)\ge B{\left(dist(z,K)\right)}^{\beta }$ if dist(z,K) ≤ 1. (LS) Here $V_K$ denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

Consider the normed space $(\mathbb{P}\left({ℂ}^N\right),|\left|·\right|\left|\right)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_g:(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|\left)\ni f\mapsto gf\in \right(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|\partial /\partial z_j{P\left|\right|\le M\left(degP\right)}^m\left|\right|P\left|\right|$, j = 1,...,N, $P\in \mathbb{P}\left({ℂ}^N\right)$, with positive constants M and m is equivalent to the inequality $\left|\right|{\partial }^N/\partial z₁...\partial z_{N}P\left|\right|\le M^{\text{'}}{\left(degP\right)}^{m^{\text{'}}}\left|\right|P\left|\right|$, $P\in \mathbb{P}\left({ℂ}^N\right)$, with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In...