An algebraic approach to the existence of valuative interpolation

TL;DR

This work characterizes the existence of valuations with prescribed values on a finite (and countable) family of ideals in an excellent regular domain of characteristic $0$ through an algebraic framework based on filtrations and the asymptotic Samuel function $\overline{\nu}$. The authors establish a central equivalence: there exists a valuation $v$ with $v(\mathfrak{a}_j)=b_j$ for all $j$ if and only if there exists a quasi-monomial valuation with the same property, which is further equivalent to $\overline{\nu}_{I_{\bullet}}(\mathfrak{a}_1\cdots\mathfrak{a}_r)=\sum_j b_j$, where $I_{\bullet}$ is the filtration generated by the scaled ideals. The finite case leverages multiplier ideals, Skoda's theorem, and asymptotic multiplier ideals to connect valuations to jump numbers, proving the main finite-interpolation theorem; the infinite case uses valuation-approximation to provide a parallel characterization and connects to complex analytic results, recovering and extending BGMY25 via an algebraic route. Overall, the paper builds a robust bridge between algebraic filtrations, asymptotic invariants, and valuation-theoretic interpolation, with implications for both commutative algebra and complex analytic geometry.

Abstract

An algebraic approach is presented for the valuative interpolation problem, which recovers and generalizes prior characterizations known in the complex analytic setting by the authors. We use the asymptotic Samuel function to give the characterization of the existence of valuative interpolation. We also give a characterization of the existence in the infinite valuative interpolation problem.

Theorems & Definitions (60)

• Theorem 1.2
• Theorem 1.3
• Example 2.1
• Example 2.2
• Lemma 2.3: see CP24
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof