Pluripotential theory on algebraic curves

TL;DR

This work develops a pluripotential framework for compact nonpolar sets on irreducible algebraic curves $A\subset \mathbb{C}^N$, connecting the Siciak–Zaharjuta extremal function $V_K$ to two families of Chebyshev constants and to transfinite diameter. It proves a plane-case uniqueness result: if $u\in L(\mathbb{C})$ with $u\le 0$ on $K$ has matching asymptotic Robin constants with $V_K^*$ along all $d$ directions, then $u=V_K^*$ outside $K$, and extends this to curves by a branch decomposition and Green-type identities. The paper then builds a directional Chebyshev theory on curves, showing that directional constants $T(K,\lambda_k)$ encode the transfinite diameter via $d(K)=\left(\prod_{k=1}^d T(K,\lambda_k)\right)^{1/d}$, and demonstrates that Chebyshev constants associated to monomial orderings reproduce these directional data. Finally, two extremal-like families $V_K^{(k)}$ and $\widetilde{V}_K^{(k)}$ are constructed whose maximum recovers $V_K^*$ on $A^0\setminus K$, providing a structured decomposition of $V_K^*$ in terms of directionally driven extremal objects and illustrated by concrete examples. The results illuminate how asymptotic direction data governs pluripotential theory on algebraic curves and offer computable representations of $V_K^*$ via extremal-like functions.

Abstract

In previous works, the second author defined directional Robin constants associated to a compact, nonpolar subset $K$ of an algebraic curve $A$ in $\mathbb{C}^N$ and related these to a natural class of Chebyshev constants for $K$. We define a second class of Chebyshev constants for $K$; relate these two classes; and utilize each of them to define two families of extremal-like functions which can be used to recover the Siciak-Zaharjuta extremal function for $K$.

Theorems & Definitions (54)

• Proposition 1.1
• Example 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Remark 2.2
• Proposition 2.3
• proof