---
title: Siciak-Zaharjuta Extremal Function
url: https://www.emergentmind.com/topics/siciak-zaharjuta-extremal-function
type: topic
---

# Siciak-Zaharjuta Extremal Function

The Siciak-Zaharjuta extremal function is a fundamental object in pluripotential theory, complex approximation, and convex geometry. It provides the maximal plurisubharmonic minorant with specified growth and boundary conditions and encodes deep geometric, analytic, and approximation-theoretic data about compact sets in complex and algebraic geometry. The extremal function is also central to the understanding of polynomial approximation, capacity theory, and the structure of equilibrium measures.

## 1. Definition and Pluripotential Characterization

Let \( K \subset \mathbb{C}^n \) be a non-pluripolar compact set. The classical Siciak–Zaharjuta (SZ) extremal function is defined by
\[
V_K(z) := \sup \left\{ \frac{1}{\deg p} \log^+ |p(z)| : p \text{ nonzero polynomial}, \ \|p\|_K \leq 1 \right\}
\]
where \( \|p\|_K = \sup_{w \in K} |p(w)| \) and \( \log^+ t = \max\{0, \log t\} \) [1908.07118, 2306.02486, 2305.08260].

An equivalent pluripotential-theoretic formulation is
\[
V_K(z) = \sup \left\{ u(z) : u \in \mathcal{L}(\mathbb{C}^n),\ u \leq 0 \text{ on } K \right\}
\]
where \( \mathcal{L}(\mathbb{C}^n) \) denotes the Lelong class: all plurisubharmonic (psh) functions with logarithmic growth at infinity, i.e., \( u(z) \leq \log^+ \|z\| + C \) for some \( C \) [1408.1756, 2306.02486].

The extremal function is plurisubharmonic, satisfies \( V_K(z) = 0 \) on \( K \) (if \( K \) is regular), and grows like \( \log^+|z| \) at infinity. It is maximal outside \( K \): \( (dd^c V_K)^n = 0 \) on \( \mathbb{C}^n \setminus K \) [1908.07118].

## 2. Polynomial Characterization, Variants, and Generalizations

For compact \( K \), the classical result states:
\[
V_K(z) = \log \Phi_K(z), \qquad
\Phi_K(z) := \sup_{m \geq 1}\; \sup\{|p(z)|^{1/m} : p \in \mathcal{P}_m(\mathbb{C}^n), \ \|p\|_K \le 1 \}
\]
where \( \mathcal{P}_m(\mathbb{C}^n) \) is the space of polynomials of degree \( \leq m \) [2305.08260].

This generalizes to weighted extremal functions and restricted exponent sets. For \( S \subset \mathbb{R}_+^n \) compact convex, define the weighted Siciak function and extremal function:
\[
V^S_{K,Q}(z) = \sup \left\{ u(z) : u \in \mathcal{L}^S, u \leq Q \text{ on } K \right\}, \qquad
\mathcal{L}^S := \{ u \in \mathrm{PSH}(\mathbb{C}^n) : u(z) \le H_S(z) + C \}
\]
with the logarithmic indicator
\[
H_S(z) := \sup_{x \in S} \langle x, \log |z| \rangle
\]
and \( Q \) an admissible weight. The function \( V^S_{K,Q} \) is the maximal \( S \)-controlled psh function subordinate to \( Q \) on \( K \); when \( S \) is the standard simplex, this reduces to the classical case [2306.02486, 2410.20370, 1911.03756].

The Siciak–Zaharjuta theorem extends to these settings: under suitable density and continuity hypotheses, \( V^S_{K,Q} = \log \Phi^S_{K,Q} \) where \( \Phi^S_{K,Q} \) is defined in terms of polynomials with exponents in integer dilates of \( S \) [2305.08260].

## 3. Geometric Decomposition and Explicit Formulae for Convex Bodies

For real convex polytopes \( K \subset \mathbb{R}^d \) of dimension \( d \), supporting simplices and strips provide an explicit decomposition:
\[
K = \bigcap_{S \in \mathcal{S}(K)} S
\]
where each \( S \) is either a simplex or a strip. The extremal function admits a max-formula [1908.07118]:
\[
V_K(z) = \max_{S \in \mathcal{S}(K)} V_S(z)
\]
For simplices \( \Delta = \mathrm{conv}(p_0, \ldots, p_d) \) with barycentric coordinates \( \lambda_j(z) \), one has [Bos–Maʻu–Waldron]:
\[
V_\Delta(z) = \log h\left( \sum_{j=0}^d |\lambda_j(z)| \right )
\]
where \( h(\eta) = \eta + \sqrt{\eta^2 - 1} \) (the inverse Joukowski map). For strips, \( V_S(z) = V_{\Delta'}(L(z)) \) under appropriate linear projections [1908.07118].

This decomposition yields finite, explicit max-formulae for \( V_K \) on polytopes, providing constructive algorithms for computation.

## 4. Foliation by Extremal Ellipses

The domain \( \mathbb{C}^d \setminus K \) is foliated by complexified ellipses \( \hat{E} \) of the form:
\[
f(\zeta) = a + c\zeta + \bar{c}/\zeta, \qquad \zeta \in \mathbb{C}^*
\]
where the real trace \( f(e^{i\theta}) \) is an inscribed real ellipse in \( K \). For every \( z \notin K \), there exists an extreme ellipse and \( \zeta_z \) with \( z = f(\zeta_z) \), and
\[
V_K(f(\zeta)) = |\log |\zeta||, \quad |\zeta| > 1
\]
These ellipses are unique and the associated leafwise extremal function is harmonic off the critical set, giving deep geometric insight [1908.07118, 1408.1756].

## 5. Regularity, Boundary Behavior, and Examples

For regular (e.g., real convex body) \( K \), \( V_K \) is continuous on \( \mathbb{C}^n \). Regularity is characterized locally: \( V_K \) is continuous at \( a \in K \) if and only if the local extremal function on a small ball intersected with \( K \) is continuous at \( a \) [2305.04171].

Hölder continuity of \( V_K \) and its weighted variants is intricately connected to Markov-type properties and the geometry of the convex set \( S \) controlling the growth. For non-pluripolar sets \( E \) with the A. Markov property and Chebyshev constant \( C(q) > 0 \), the log of the extremal function is Hölder continuous [1808.01604]. However, for general convex (but non-lower set) \( S \), weighted extremals may fail to be Hölder continuous [2410.20370].

Explicit examples:
- For \( K = [-1,1]^d \), \( V_K(z) = \max_j \log h(z_j) \).
- For the standard simplex, Baran's formula: \( V_\Sigma(z) = \log h \left(\sum_{j=1}^d |z_j| + |1 - \sum z_j|\right) \) [1908.07118].

## 6. Extensions: Toric, Kähler, and Singular Settings

On toric varieties \( X = X_\Sigma \) with ample line bundle \( L \), the Siciak theorem generalizes:
\[
V_{K,L}(x) = \sup \left\{ \frac{1}{d} \log |s(x)|_{h_0^d} : s \in H^0(X, L^d), \sup_K |s|_{h_0^d} \leq 1 \right\}
\]
with appropriate reference metrics and polytopal data encoding the growth [1101.2911].

On compact Kähler manifolds, for the extremal function \( V_K \) defined via ω-psh functions, continuity and Hölder continuity are local properties, reducing to classical results via holomorphic charts [2305.04171].

On singular varieties, \( V_{K,X} \) admits characterization as the envelope of Poisson or disc functionals, extending Lempert’s and Lárusson–Sigurdsson’s extremal disc formulae under local irreducibility [1109.3947].

## 7. Applications and Connections

The Siciak–Zaharjuta extremal function is instrumental in:
- Analysis of polynomial and holomorphic approximation, via the Bernstein–Walsh–Siciak theorems [2306.02486, 1801.02401].
- Equilibrium (Monge–Ampère) measures, Robin functions, and equidistribution of Fekete points [1408.1756, 2305.04171].
- Convexity properties and Markov-type inequalities for derivatives of polynomials [1808.01604].
- Pluripotential theory on algebraic curves, including the computation of directional Robin constants and Chebyshev constants [2601.07639].
- Nonarchimedean potential theory, via analogues for Berkovich spaces [2107.03539].
- Noncommutative analogues of the extremal function and emergent free probability theory [2106.06212].

Recent work provides explicit computability for convex polytopes and bodies, clarifies the obstacle to Hölder continuity for generalized extremals, and systematically generalizes Siciak-type theory to weighted, toric, and algebraic curve contexts. The function thus serves as a bridge connecting pluripotential theory, convex geometry, algebraic geometry, and analytic approximation.

Source: https://www.emergentmind.com/topics/siciak-zaharjuta-extremal-function