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The Bernstein-Sato $b$-function of the Vandermonde determinant

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Asilata Bapat, Robin Walters

Algebraic Geometry · Submitted 3 Mar 2015

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

The authors' main claim This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case.

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arXiv ID1503.01055 (version 1)
Pith IDZOUCGWK5, Pith's permanent record of this paper and its review
CitationAsilata Bapat, Robin Walters. "The Bernstein-Sato $b$-function of the Vandermonde determinant." arXiv:1503.01055 (2015). https://pith.science/paper/ZOUCGWK5
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,040 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.76 · compares with prior results 0.60 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the Vandermonde determinant $\xi$. We use a result of Opdam to produce a lower bound for the $b$-function of $\xi$. This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case. In our second set of results, we show the duality of two $\mathcal{D}$-modules, and conclude that the roots of the $b$-function of $\xi$ are symmetric about $-1$. We then use some results about jumping coefficients to prove an upper bound for the $b$-function of $\xi$, and finally we conjecture a formula for the $b$-function of $\xi$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. background 1.00 0.04 0.02 asserted 1.00
s1 We describe a few different results towards computing the $b$-function of the Vandermonde determinant $\xi$. contribution 0.99 0.86 0.09 asserted 0.93
s2 We use a result of Opdam to produce a lower bound for the $b$-function of $\xi$. method 0.68 0.87 0.07 asserted 1.00
s3 This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case. interpretation 0.74 0.82 0.35 asserted 1.00
s4 In our second set of results, we show the duality of two $\mathcal{D}$-modules, and conclude that the roots of the $b$-function of $\xi$ are symmetric about $-1$. result 0.86 0.95 0.16 asserted 1.00
s5 We then use some results about jumping coefficients to prove an upper bound for the $b$-function of $\xi$, and finally we conjecture a formula for the $b$-function of $\xi$. result 0.53 0.94 0.28 speculative 0.64

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

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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

The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the Vandermonde determinant $\xi$. We use a result of Opdam to produce a lower bound for the $b$-function of $\xi$. This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case. In our second set of results, we show the duality of two $\mathcal{D}$-modules, and conclude that the roots of the $b$-function of $\xi$ are symmetric about $-1$. We then use some results about jumping coefficients to prove an upper bound for the $b$-function of $\xi$, and finally we conjecture a formula for the $b$-function of $\xi$.

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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

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Uses data or values from 1 paper
  • By Mustaţă [ref] , the minimal jumping coefficient of $𝒜$ is [equation] Using the above relationship between $b$-functions and jumping coefficients, we can rephrase this (as was done in [ref] ) as follows: [equation]

    Bernstein-Sato polynomials of hyperplane arrangements · Partial proof of main conjecture · 0.81

Cites as support 1 paper
Cites as background 1 paper
  • This active area of research draws together the study of $b$-functions with several other important singularity invariants, including Igusa zeta functions, jumping coefficients, and local monodromy [ref] .

    Bernstein-Sato polynomials of hyperplane arrangements · Introduction · 0.99

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