We consider the class of all homogeneous, possibly non-reduced, polynomials f whose associated reduced projective divisor D_red⊂ℙ^n-1 has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables n and with d denoting the degree of f, we characterize when -n/d is a root of the Bernstein–Sato polynomial of f in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense.

Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings. In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. This variety is typically still not smooth, but it is the closure of a smooth subvariety of a torus. The latter then a smooth, tropical compactification, using work of Tevelev. We construct explicitly such a compactification and a morphism to the normalized Nash blow-up for every configuration, described in terms of bipermutohedral matroid combinatorics introduced by Ardila, Denham and Huh. Along the way, we find that the normalized Nash blow-up of the configuration hypersurface has strongly F-regular singularities in positive characteristic. We deduce this by certifying F-rationality of its biprojective cone, and infer from it that the normalized Nash blow-up has rational singularities over the complex numbers.

We consider the Bernstein-Sato polynomial of a locally quasi-homogeneous polynomial f is an element of R = C[x1, x2, x3]. We construct, in the analytic category, a complex of DX[s]-modules that can be used to compute the DX[s]-dual of DX[s]fs-1 as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narvaez Macarro where a freeness assumption was required. We derive many results about the zeros of the Bernstein-Sato polynomial. First, we prove each nonvanishing degree of the zeroth local cohomology of the Milnor algebra H0m(R/(partial derivative f)) contributes a root to the Bernstein-Sato polynomial, generalizing a result of M. Saito (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeros of the Bernstein-Sato polynomial admit a partial symmetry about -1, extending a result of Narvaez Macarro that again required freeness. We give applications to very small roots, the twisted logarithmic comparison theorem, and more precise statements when f is additionally assumed to be homogeneous. Finally, when f defines a hyperplane arrangement in C3 we give a complete formula for the zeros of the Bernstein-Sato polynomial of f. We show all zeros except the candidate root -2 + (2/ deg(f)) are (easily) combinatorially given; we give many equivalent characterizations of when the only noncombinatorial candidate root -2 + (2/ deg(f)) is in fact a zero of the Bernstein-Sato polynomial. One equivalent condition is the non-vanishing of H0m(R/(partial derivative f))deg(f)-1.

We study the canonical Hodge filtration on the sheaf 𝒪_X(*D) of meromorphic functions along a divisor. For a germ of an analytic function f whose Bernstein-Sato's polynomial's roots are contained in (-2,0), we: give a simple algebraic formula for the zeroeth piece of the Hodge filtration; bound the first step of the Hodge filtration containing f^-1. If we additionally require f to be Euler homogeneous and parametrically prime, then we extend our algebraic formula to compute every piece of the canonical Hodge filtration, proving in turn that the Hodge filtration is contained in the induced order filtration. Finally, we compute the Hodge filtration in many examples and identify several large classes of divisors realizing our theorems.

For a rank 1 local system on the complement of a reduced divisor on a complex manifold X X , its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we study necessary or sufficient conditions for a quasi-isomorphism from its twisted logarithmic subcomplex, called the logarithmic comparison theorem (LCT), by using a stronger version in terms of the associated complex of D X D_X -modules. In case the connection is a pullback by a defining function f f of the divisor and the residue is α \alpha , we prove among others that if LCT holds, the annihilator of f α − 1 f^{\alpha -1} in D X D_X is generated by first order differential operators and α − 1 − j \alpha -1-j is not a root of the Bernstein-Sato polynomial for any positive integer j j . The converse holds assuming either of the two conditions in case the associated complex of D X D_X -modules is acyclic except for the top degree. In the case where the local system is constant, the divisor is defined by a homogeneous polynomial, and the associated projective hypersurface has only weighted homogeneous isolated singularities, we show that LCT is equivalent to that − 1 -1 is the unique integral root of the Bernstein-Sato polynomial. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues, which is an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity. Here the zero-extension case is also treated.

. For a reduced hyperplane arrangement we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao and Yuzvinsky. We also prove that: every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite dimensional lin- ear algebra. Finally, we give some D X -module applications: for example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in ( − 2 + 1 /d, 0) .

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standard Bernstein-Sato ideals for reduced, generic arrangements: we compute the Bernstein-Sato ideal for a factorization into linear forms and we compute its zero locus for other factorizations.

We present a variant of the Peskine–Szpiro Acyclicity Lemma, and hence a way to certify exactness of a complex of finite modules over a large class of (possibly) noncommutative rings. Specifically, over the class of Auslander regular rings. In the case of relative 𝒟_X-modules, for example 𝒟_X[s_1, …, s_r]-modules, the hypotheses have geometric realizations making them easier to authenticate. We demonstrate the efficacy of this lemma and its various forms by independently recovering some results related to Bernstein–Sato polynomials.

Given a complex germ $f$ near the point $\mathfrak{x}$ of the complex manifold $X$, equipped with a factorization $f = f_{1} \cdots f_{r}$, we consider the $\mathscr{D}_{X,\mathfrak{x}}[s_{1}, \dots, s_{r}]$-module generated by $ F^{S} := f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$. We show for a large class of germs that the annihilator of $F^{S}$ is generated by derivations and this property does not depend on the chosen factorization of $f$. We further study the relationship between the Bernstein-Sato variety attached to $F$ and the cohomology support loci of $f$, via the $\mathscr{D}_{X,\mathfrak{x}}$-map $\nabla_{A}$. This is related to multiplication by $f$ on certain quotient modules. We show that for our class of divisors the injectivity of $\nabla_{A}$ implies its surjectivity. Restricting to reduced, free divisors, we also show the reverse, using the theory of Lie-Rinehart algebras. In particular, we analyze the dual of $\nabla_{A}$ using techniques pioneered by Narvaez-Macarro. As an application of our results we establish a conjecture of Budur in the tame case: if $\text{V}(f)$ is a central, essential, indecomposable, and tame hyperplane arrangement, then the Bernstein-Sato variety associated to $F$ contains a certain hyperplane. By the work of Budur, this verifies the Topological Mulivariable Strong Monodromy Conjecture for tame arrangements. Finally, in the reduced and free case, we characterize local systems outside the cohomology support loci of $f$ near $\mathfrak{x}$ in terms of the simplicity of modules derived from $F^{S}.$

For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying the functional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in \text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdots f_{r}^{s_{r} + 1}.$ Generalizing techniques due to Maisonobe, we compute the zero locus of the standard Bernstein--Sato ideal in the sense of Budur (i.e. $f^{\prime} = 1)$ for any factorization of a free and reduced $f$ and for certain factorizations of a non-reduced $f$. We also compute the roots of the Bernstein--Sato polynomial for any power of a free and reduced arrangement. If $f$ is tame, we give a combinatorial formula for the roots lying in $[-1,0).$ For $f^{\prime} \neq 1$ and any factorization of a line arrangement, we compute the zero locus of this ideal. For free and reduced arrangements of larger rank, we compute the zero locus provided $\text{deg}(f^{\prime}) \leq 4$ and give good estimates otherwise. Along the way we generalize a duality formula for $\mathscr{D}_{X,\mathfrak{x}}[S]f^{\prime}f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$ that was first proved by Narvaez-Macarro for $f$ reduced, $f^{\prime} = 1$, and $r = 1.$ As an application, we investigate the minimum number of hyperplanes one must add to a tame $f$ so that the resulting arrangement is free. This notion of freeing a divisor has been explicitly studied by Mond and Schulze, albeit not for hyperplane arrangements. We show that small roots of the Bernstein--Sato polynomial of $f$ can force lower bounds for this number.