{"paper":{"title":"The Bernstein-Sato $b$-function of the Vandermonde determinant","arxiv_id":"1503.01055","primary_cat":"math.AG","submitted_at":"2015-03-03T18:53:57+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"contribution","text":"We describe a few different results towards computing the $b$-function of the Vandermonde determinant $\\xi$.","hedge":{"p":0.93,"label":"asserted"},"own_p":0.86,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":2,"p":0.68,"id":"s2","own":true,"role":"method","text":"We use a result of Opdam to produce a lower bound for the $b$-function of $\\xi$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.57,"role_probabilities":{"gap":0.0,"method":0.68,"result":0.06,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.26,"interpretation":0.0}},{"i":3,"p":0.74,"id":"s3","own":true,"role":"interpretation","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.82,"main_claim_p":0.35,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.26,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.74}},{"i":4,"p":0.86,"id":"s4","own":true,"role":"result","text":"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 $-1$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.16,"quantitative":true,"quantitative_p":0.52,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.86,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}},{"i":5,"p":0.53,"id":"s5","own":true,"role":"result","text":"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 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