Improved Bounds for the s-multiplicity

TL;DR

This work advances the theory of $s$-multiplicity and the $h$-function in prime characteristic by deriving explicit formulas for their behavior under two fundamental ring constructions: fiber products $R \times_T S$ and idealizations $R \ltimes M$. The authors establish core equalities such as $h_s(R \times_T S)=h_s(R)+h_s(S)-h_s(T)$ in the appropriate dimension regimes and show corresponding additivity for $e_s$, then translate these into bounds and comparisons for $s$-multiplicity in idealizations. Building on these formulas, the paper proves new Taylor-Miller-type lower bounds and proves the $s$-analogue of the Watanabe-Yoshida conjecture for these constructions, extending known results to broader classes of rings (e.g., Cohen-Macaulay and non-regular complete intersections). The results unify and extend the understanding of how $s$-multiplicity behaves under standard ring operations, with implications for singularity theory and interpolation between Hilbert-Kunz and Hilbert-Samuel frameworks. Overall, the work broadens the toolkit for estimating $s$-multiplicity in composite rings and strengthens connections to longstanding conjectures in positive characteristic.

Abstract

Let (RmR), (SmS) and (TmT) be Noetherian local rings sharing the same residue eld k and prime characteristic p > 0. We establish some formulas relating the h-function and s-multiplicity of the ber product R T S in terms of the h-functions and s-multiplicities of R, T and S. Furthermore, we derive formulas that connect the h-function and s-multiplicity of the idealization ring R M to the corresponding invariants of R and M, where M is a nitely generated R-module. As applications of these results, we derive new estimates for the Taylor-Miller question and the Watanabe-Yoshida conjecture concerning s-multiplicity.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• proof
• Corollary 3.2