Valuative Interpolation: A Theoretical Framework

• Valuative interpolation is a mathematical framework that constructs valuations on local rings by interpolating prescribed function values, uniting algebraic and analytic methods.
• It leverages analytic criteria using plurisubharmonic weights, including Zhou weights and Tian functions, to extend classical valuation techniques into singular analytic settings.
• Its methodology applies to polynomial and real-analytic rings, offering a unified approach to address challenges in measuring singularity orders and integrability in complex geometry.

Valuative interpolation is a mathematical framework focused on the existence and construction of valuations on local rings (typically of analytic, polynomial, or real-analytic functions at a point) that interpolate prescribed values on a finite set of elements. Central to this theory are analytic criteria phrased in terms of relative types of plurisubharmonic (psh) weights, and novel mechanisms—Zhou weights and Tian functions—which extend classical tools from complex geometry to singular analytic settings. The problem, and its solutions, unify valuation theory, pluripotential theory, and singularity analysis at the interface of algebraic and analytic geometry (Bao et al., 25 Oct 2025, Bao et al., 4 Jan 2026).

1. Formal Definition and Core Problem

Let $\mathcal{O}$ denote a local $\mathbb{C}$–algebra, typically $\mathcal{O}_{X,o}$ for an irreducible complex analytic germ $(X,o)\subset(\mathbb{C}^n,o)$, or its weakly holomorphic extension $\mathcal{O}^w_{X,o}$. A valuation on $\mathcal{O}$ is a map $$\nu:\mathcal{O}\setminus\{0\}\rightarrow\mathbb{R}_{\geq0}$$ satisfying:

• $\nu(fg)=\nu(f)+\nu(g)$ (multiplicativity),
• $\nu(f+g)\geq\min\{\nu(f),\nu(g)\}$ (ultrametric additivity),
• $\nu(\text{const})=0$.

Given nonzero elements $f_1,\ldots,f_m\in\mathcal{O}$ and target values $a_1,\ldots,a_m\geq0$, valuative interpolation asks: Does there exist a valuation $\nu$ on $\mathcal{O}$ such that $\nu(f_j)=a_j$ for all $j$? This generalizes classical problems of specifying orders of vanishing or singularities at given loci (Bao et al., 25 Oct 2025).

Equivalently, one seeks an $\mathbb{R}$-linear functional $\mathcal{I}:\mathcal{O}\rightarrow\mathbb{R}$, subject to the valuation axioms, matching interpolation data on a finite set.

2. Analytic Criterion: Relative Type and Main Theorems

The existence of a solution is governed by explicit analytic tests involving plurisubharmonic functions. With $f_0=1$, $F=\prod_{j=0}^m f_j$, and $\varphi=\log(\sum_{j=1}^m|f_j|^{1/a_j})$:

• Main equivalence [Theorem 1.3, (Bao et al., 4 Jan 2026)]: There exists $\nu$ with $\nu(f_j)=a_j$ for all $j$ iff

$\sigma(\log|F|,\varphi) = \sum_{j=1}^m a_j$

where $\sigma(\psi,\varphi)$ denotes the relative type of a psh function $\psi$ with respect to $\varphi$:

$$\sigma(\psi,\varphi)=\sup\{c\geq0 : \psi\leq c\,\varphi+O(1)\text{ near }o\}$$

This criterion applies uniformly in the holomorphic, weakly holomorphic, and (through extensions) quotient settings associated to polynomial and real-analytic rings (Bao et al., 25 Oct 2025, Bao et al., 4 Jan 2026).

3. Zhou Weights and Tian Functions: Structure and Mechanisms

Near singular points, classical maximal weights are insufficient; Zhou weights generalize these to singular germs $(X,o)$:

• A psh $\Phi$ on $X$ is a Zhou weight for a reference datum $(f_0,\varphi_0)$ if
  1. For large $N$, $\Phi+N\log|z|$ yields $\left|f_0\right|^2 e^{-2\varphi_0-2\Phi-2N\log|z|}$ integrable,
  2. $\left|f_0\right|^2 e^{-2\varphi_0-2\Phi}$ fails to be integrable at $o$,
  3. Minimality: If $\varphi'\geq\Phi+O(1)$ also fails integrability, then $\varphi'=\Phi+O(1)$.

Existence is guaranteed via the Strong Openness Theorem.

Defining $\nu(f):=\sigma(\log|f|,\Phi_{o,\max})$ for such Zhou weight $\Phi_{o,\max}$ yields a genuine valuation, called a Zhou valuation (Bao et al., 25 Oct 2025, Bao et al., 4 Jan 2026).

Tian functions generalize relative-type via mixed orders:

$$\text{Tn}(t) = \sup\left\{c : |f_0|^2\,e^{-2\varphi_0}\,e^{2t\psi}\,e^{-2c\Phi_{o,\max}}\text{ is }L^1\text{ near }o\right\}$$

$\text{Tn}(t)$ is concave in $t$, and $\text{Tn}'(0)=\sigma(\psi,\Phi_{o,\max})$ yields Zhou valuations as derivatives of Tian functions.

Structure	Object	Key Properties / Usage
Analytic type	Relative type $\sigma$	Measures comparative singularity for interpolation
Weight function	Zhou weight $\Phi$	Minimal psh controlling integrability
Associated map	Tian function $\text{Tn}$	Encodes valuation via derivative at $t=0$

4. Extension to Polynomial and Real-analytic Quotients

Valuative interpolation extends to quotient rings via embeddings and explicit analytic criteria (Bao et al., 25 Oct 2025, Bao et al., 4 Jan 2026):

• Complex polynomial rings: For $I\subset\mathbb{C}[z_1,\ldots,z_n]$ prime, $X=V(I)$ irreducible, $\mathbb{C}[z]/I\subset\mathcal{O}_{X,o}$, similar criteria hold. If $\bigcap_{j=1}^m\{f_j=0\}=\{o\}$, the necessary and sufficient criterion reduces to

$\sigma(\log|F|,\varphi)=\sum_j a_j$

and every valuation centered at $o$ matches this data.

• Real-analytic and real-polynomial cases: Analogous results hold via complexification embedding, provided primality persists under complexification. In the real-polynomial case, the converse also requires that $\nu(x_i)>0$ for all $i$, and the conditions become equivalent when the common zero set is $\{o\}$.

5. Illustrative Example: The Plane Cusp

Consider $I=(z_1^3 - z_2^2)\subset\mathbb{C}[z_1, z_2]$; $X=V(I)$ is the standard cusp. Parametrize by $t\mapsto(z_1,z_2)=(t^2,t^3)$:

• The unique valuation is

$$\nu: \mathbb{C}[z_1,z_2]/I \rightarrow \mathbb{Z}_{\geq0},\ \nu(f(t^2,t^3)) = \operatorname{ord}_{t=0}(f(t^2,t^3))$$

• For $f_1=[z_1]$, $f_2=[z_2]$, $a_1=2$, $a_2=3$:

$$F=z_1^2z_2^3,\ \varphi=\log(|t|^2 + |t|^3) = 2\log|t| + O(1),\ \log|F| = 7\log|t| + O(1)$$

The correct weighting ensures $\sigma(\log|F|,\varphi)=a_1+a_2=5$, precisely recovering prescribed orders via the criterion.

6. Proof Strategy and Technical Aspects

The existence of Zhou weights leverages the strong openness property of multiplier ideals. Every Zhou weight defines a valuation $\nu(f)=\sigma(\log|f|,\Phi_{\max})$.

For necessity, if $\nu(f_j)=a_j$, then $\sigma(\log|F|,\varphi)=\sum a_j$ via subadditivity. For sufficiency, Tian functions and concavity arguments, together with Skoda-type division, are deployed to construct approximate weights whose associated valuations satisfy the prescribed interpolation values. A diagonal subsequence argument using Noetherianity yields a limiting valuation.

Technical challenges include extension of Choquet's lemma and Ohsawa–Takegoshi integrability to singular settings (requiring resolution of singularities) and control of Tian function asymptotics.

7. Connections, Extensions, and Characterizations

• Every valuation with suitable finiteness properties (bounded asymptotic LCT jump) can be realized as a relative type with respect to a tame maximal weight (Bao et al., 25 Oct 2025).
• Extension of the Boucksom–Favre–Jonsson theorem establishes comparable inclusion properties using only quasimonomial Zhou valuations.
• Differentiability of Tian functions at $t=0$ acts as a selector for valuations arising from tame maximal weights; the derivative yields the valuation data (Bao et al., 25 Oct 2025).

A plausible implication is that the analytic tools summarized by Zhou weights and Tian functions provide a flexible bridge between the algebraic world of valuations and the analytic world of singularities, suggesting broader applicability in complex and real singularity theory.

• Bao, Guan, Mi, Yuan: "The existence of valuative interpolation" (Bao et al., 25 Oct 2025).
• Bao, Guan, Mi, Yuan: "The existence of valuative interpolation at a singular point" (Bao et al., 4 Jan 2026).