Generators and Support Loci of Bernstein–Sato Ideals

Definitions and mathematical statement

The supplied record concerns a germ \[ F=(f_1,\ldots,f_r):({\mathbb{C}}^n,0)\longrightarrow({\mathbb{C}}^r,0) \] and an associated Bernstein--Sato ideal denoted by $B_F$. The record does not supply the defining functional equation for this ideal, the precise class of germs, or the definition of the cohomology-support locus $V_F$. These omissions prevent an unconditional, fully specified formulation.

Writing $R={\mathbb{C}}[s_1,\ldots,s_r]$ for the polynomial ring in the parameter variables, the first proposed assertion is that, for every tuple $F$ in the intended class, the ideal $B_F\subseteq R$ admits a generating family $(g_\lambda)_{\lambda\in\Lambda}$ whose members are finite products of affine linear factors of the form \[ \ell(s_1,\ldots,s_r)=a_1s_1+\cdots+a_rs_r+b, \qquad a_i\in{\mathbb{Q}}_{\geq0},\quad b\in{\mathbb{Q}}_{>0}. \] Thus the proposed assertion is ideal generation, $B_F=(g_\lambda\mid\lambda\in\Lambda)$, and not merely a description of its zero set. The supplied coefficient conditions do not explicitly require any $a_i$ to be nonzero.

For an ideal $I\subseteq R$, define its complex zero set by \[ Z(I)=\{s\in{\mathbb{C}}^r:p(s)=0\text{ for every }p\in I\}. \] With the explicitly provisional convention \[ \operatorname{Exp}(s_1,\ldots,s_r) =\left(e^{2\pi\mathrm{i}s_1},\ldots,e^{2\pi\mathrm{i}s_r}\right) \in({\mathbb{C}}^\times)^r, \] the second proposed assertion is \[ \operatorname{Exp}\left(Z(B_F)\right)=V_F. \] The source extraction does not specify the exponential convention or define $V_F$, including the spaces, local systems, cohomological degrees, and points involved in its construction. Consequently this equality records the supplied target but remains unresolved as a precise mathematical statement. In particular, no choice between support at the origin and support obtained from nearby points is made here. The intended definition of $B_F$, including its parameter-shift convention, must also be recovered before either assertion can be assessed.

Short English statement

For every germ tuple $F$ in the intended class, the problem asks whether its Bernstein--Sato ideal has generators that are products of affine linear factors with nonnegative rational slopes and positive rational constant terms, and whether exponentiating its zero set gives the stated cohomology-support locus. The supplied extraction leaves essential definitions unspecified, so this is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003227$, title Generators and Support Loci of Bernstein--Sato Ideals, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. No proposer is identified; the supplied proposed year is 2016.
• Supplied report citation: Singularities (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/46, pp. 2625--2691. https://doi.org/10.4171/owr/2016/46. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. Its literature-source list also labels this reference Oberwolfach Report 46.
• Supplied literature reference: Budur, Bernstein-Sato ideals and local systems. https://doi.org/10.5802/aif.2939.
• Supplied literature assessment dated exactly 2026-08-21: the record is described as an incomplete extraction of the multivariable Bernstein--Sato conjecture, whose full assertion therefore cannot be certified as solved; the unresolved generator statement is treated as open. This is supplied dated curation derived from the Oberwolfach material, not an independently verified literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the problem open and the statement corrected_verified, and reports that the clean statement was checked against the official report. These are supplied registry qualifications, not verification performed for this draft. The supplied report title uses 2017, while the DOI and proposed-year metadata use 2016; both are retained without resolving their relationship.