Siciak-Zaharjuta Extremal Function

• The Siciak-Zaharjuta extremal function is a key concept in pluripotential theory, defined as the maximal plurisubharmonic minorant with logarithmic growth constraints on complex spaces.
• It encapsulates deep geometric, analytic, and approximation data, facilitating polynomial and holomorphic approximations for compact sets in complex and algebraic geometry.
• Its extensions to weighted, toric, and singular settings broaden applications across equilibrium measure analysis, convex geometry, and capacity theory.

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 $\log^+ t = \max\{0, \log t\}$ (Ma`u, 2019, Magnússon et al., 2023, Magnússon et al., 2023).

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 $C$ (Burns et al., 2014, Magnússon et al., 2023).

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 $\mathbb{C}^n \setminus K$ (Ma`u, 2019).

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 $\leq m$ (Magnússon et al., 2023).

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 (Magnússon et al., 2023, Snorrason, 2024, Bayraktar et al., 2019).

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 $S$ (Magnússon et al., 2023).

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 (Ma`u, 2019): $V_K(z) = \max_{S \in \mathcal{S}(K)} V_S(z)$ 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 (Ma`u, 2019).

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 (Ma`u, 2019, Burns et al., 2014).

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 $a$ (Nguyen, 2023).

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 (Baran et al., 2018). However, for general convex (but non-lower set) $S$, weighted extremals may fail to be Hölder continuous (Snorrason, 2024).

Explicit examples:

• For $K = [-1,1]^d$, $V_K(z) = \max_j \log h(z_j)$.
• For the standard simplex, Baran's formula: $$V_\Sigma(z) = \log h \left(\sum_{j=1}^d |z_j| + |1 - \sum z_j|\right)$$ (Ma`u, 2019).

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 (Branker et al., 2011).

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 (Nguyen, 2023).

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 (Drnovsek et al., 2011).

7. Applications and Connections

The Siciak–Zaharjuta extremal function is instrumental in:

• Analysis of polynomial and holomorphic approximation, via the Bernstein–Walsh–Siciak theorems (Magnússon et al., 2023, Bloom et al., 2018).
• Equilibrium (Monge–Ampère) measures, Robin functions, and equidistribution of Fekete points (Burns et al., 2014, Nguyen, 2023).
• Convexity properties and Markov-type inequalities for derivatives of polynomials (Baran et al., 2018).
• Pluripotential theory on algebraic curves, including the computation of directional Robin constants and Chebyshev constants (Levenberg et al., 12 Jan 2026).
• Nonarchimedean potential theory, via analogues for Berkovich spaces (Stawiska, 2021).
• Noncommutative analogues of the extremal function and emergent free probability theory (Belinschi et al., 2021).

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.