The existence of valuative interpolation

TL;DR

This paper establishes necessary and sufficient conditions for the existence of valuative interpolations on rings of germs of functions at the origin, using advanced valuation tools and exploring their properties and relationships.

Contribution

It introduces new criteria for valuative interpolations and extends known results to Zhou valuations and their connection with Tian functions.

Findings

Criteria for valuative interpolations established

Extension of Boucksom--Favre--Jonsson results to Zhou valuations

Relationship between Zhou valuations and Tian functions' differentiable points

Abstract

In this article, using key tools including Zhou valuations, Tian functions and a convergence result for relative types, we establish necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and real analytic functions at the origin in $\mathbb{C}^{n}$ and $\mathbb{R}^{n}$, respectively. For the cases of polynomial rings with complex and real coefficients, we establish separate necessary conditions and sufficient conditions, which become both necessary and sufficient when the intersection of the zero sets of the given polynomials is the set of the origin in $\mathbb{C}^{n}$. Furthermore, we obtain a necessary and sufficient condition for a valuation to be of the form given by a relative type with respect to a tame maximal weight. We demonstrate a result of Boucksom--Favre--Jonsson on quasimonomial valuations also holds for quasimonomial Zhou valuations. Finally, we obtain a relationship between Zhou valuations and the differentiable points of Tian functions.