On Lower Bounds for $s$-multiplicities

Lance Edward Miller; William D. Taylor

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On Lower Bounds for $s$-multiplicities

Commutative Algebra · Submitted 21 Aug 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,591 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.41 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.43 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities.	background 1.00	0.04	0.03	asserted 1.00
s1	The obvious goal is to use this as a tool for deforming results from one to the other.	background 0.83	0.05	0.03	asserted 0.87
s2	The values in this family which do not match these classic variants however are not known yet to be well-behaved.	gap 1.00	0.06	0.04	asserted 0.88
s3	This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.	contribution 1.00	0.92	0.47	asserted 0.72

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 10 sentences without a confident label

Uses the method or tool of 1 paper

As $B$ is toric, [ref] gives $e_s(^n)=(U)/_s(d)$, where [equation] By direct computation [equation]

Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity · Estimations of $s$-multiplicity · 0.86

Cites as background 1 paper

The limit of colengths $h_s(I,J) := lim_e→∞ λ( R / e )/p^ed$ exists [ref] .

Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity · A brief review of $s$-multiplicity · 0.86

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