An algebraic approach to the existence of valuative interpolation

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.

💡 Research Summary

The paper develops a purely algebraic framework for the valuative interpolation problem, which asks whether there exists a valuation (v) on a Noetherian domain (R) such that prescribed values (b_1,\dots,b_r\ge0) are attained on given non‑zero ideals (\mathfrak a_1,\dots,\mathfrak a_r). The authors work in the setting of an excellent regular domain of equicharacteristic zero, but the main results hold for any Noetherian integral domain.

The central tool is the asymptotic Samuel function associated to a filtration of ideals. For a filtration (\mathfrak J_\bullet={\mathfrak J_m}{m\ge0}) the order of an ideal (\mathfrak a) is (\nu{\mathfrak J_\bullet}(\mathfrak a)=\sup{m\mid \mathfrak a\subseteq\mathfrak J_m}). The asymptotic Samuel function is then defined by (\nu_{\mathfrak J_\bullet}(\mathfrak a)=\lim_{m\to\infty}\nu_{\mathfrak J_\bullet}(\mathfrak a^m)/m). Basic properties (monotonicity, scaling, subadditivity, and a weak multiplicativity) are proved, mirroring classical results of Samuel.

Given the data ((\mathfrak a_j,b_j)), the authors introduce the filtration
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