2019/09/02 by Daniel Bath, Bath, Daniel
Computer Science · Engineering · Mathematics · #14F10 (Primary) 32S40 #32C38 (Secondary) #32S05 #32S22 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1909.00547
openalex publication_date 2019/09/02 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
For a central, not necessarily reduced, hyperplane arrangement f equipped\nwith any factorization f = f1 \⋯ fr and for f\′ dividing\nf, we consider a more general type of Bernstein--Sato ideal consisting of the\npolynomials B(S) \∈ \ℂ[s1, \…, sr] satisfying the\nfunctional equation B(S) f\′ f1^s1 \⋯ fr^sr \∈\n\An(\ℂ)[s1, \…, sr] f1^s1 + 1 \⋯\nfr^sr + 1.\n Generalizing techniques due to Maisonobe, we compute the zero locus of the\nstandard Bernstein--Sato ideal in the sense of Budur (i.e. f\′ = 1)\nfor any factorization of a free and reduced f and for certain factorizations\nof a non-reduced f. We also compute the roots of the Bernstein--Sato\npolynomial for any power of a free and reduced arrangement. If f is tame, we\ngive a combinatorial formula for the roots lying in [-1,0).\n For f\′ \≠ 1 and any factorization of a line arrangement, we\ncompute the zero locus of this ideal. For free and reduced arrangements of\nlarger rank, we compute the zero locus provided deg(f\′) \≤ 4\nand give good estimates otherwise. Along the way we generalize a duality\nformula for mathscrD_X, mathfrakx[S]f\′f1^s1 \⋯\nfr^sr that was first proved by Narv 'aez-Macarro for f reduced,\nf\′ = 1, and r = 1.\n As an application, we investigate the minimum number of hyperplanes one must\nadd to a tame f so that the resulting arrangement is free. This notion of\nfreeing a divisor has been explicitly studied by Mond and Schulze, albeit not\nfor hyperplane arrangements. We show that small roots of the Bernstein--Sato\npolynomial of f can force lower bounds for this number.\n