Uniqueness in Linear Programs for Binomial Multiplier Ideals

Definitions and mathematical statement

Let $k$ be a field of characteristic zero, and let $\mathfrak a\subseteq k[x_1,\ldots,x_n]$ be a binomial ideal containing no nonzero monomial. Here $x^c=\prod_{j=1}^n x_j^{c_j}$ for $c\in\mathbb Z_{\geq0}^n$. Let \[ \mathfrak a=(f_1,\ldots,f_r),\qquad f_i=x^{a_i}-\gamma_i x^{b_i},\qquad a_i,b_i\in\mathbb Z_{\geq0}^n,\quad \gamma_i\in k^\times, \] be any minimal binomial generating system, meaning that no proper subcollection generates $\mathfrak a$. Regard the exponent vectors as columns and define \[ A=\begin{pmatrix} a_1&\cdots&a_r&b_1&\cdots&b_r\\ &I_r&&&I_r& \end{pmatrix}\in\mathbb Z_{\geq0}^{(n+r)\times 2r}, \] where the bottom block is $(I_r\ I_r)$. For $z=(\mu_1,\ldots,\mu_r,\nu_1,\ldots,\nu_r)^{\mathsf T}$, put \[ P=\{z\in\mathbb Q_{\geq0}^{2r}:Az\leq\mathbf1_{n+r}\},\qquad L(z)=\sum_{i=1}^r(\mu_i+\nu_i), \] with the inequality interpreted coordinatewise. Define the set of optimal rational solutions by \[ \mathcal O=\{z\in P:L(z)\geq L(w)\text{ for every }w\in P\}. \] The question is whether, for every such field, ideal, and minimal binomial generating system, \[ \exists z\in\mathcal O\quad\forall z'\in\mathcal O,\qquad z'\ne z\ \Longrightarrow\ Az'\ne Az. \] Thus the requested solution must be the only optimal rational solution having its particular image under $A$. The supplied original formulation refers to the linear program of Proposition 5; that proposition is not reproduced in the record. The matrix and rational domains above follow the supplied clean statement, and their correspondence with that proposition remains subject to source review.

Short English statement

For every binomial ideal containing no nonzero monomial over a field of characteristic zero, and every minimal binomial generating system, form the stated rational linear program from the exponent vectors. Must some optimal solution be the only optimal rational solution with its image under $A$? The supplied record labels this question open; that status is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001234$, exact source problem code catalogue entry, titled "Uniqueness in Linear Programs for Binomial Multiplier Ideals." The record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2009; no individual proposer is identified.
• Multiplier Ideal Sheaves in Algebraic and Complex Geometry (2009), Oberwolfach Reports, pp. 1101--1156, DOI 10.4171/owr/2009/21: https://doi.org/10.4171/owr/2009/21. This is the supplied source report and the literature source labeled "original report." The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Shibuta Takagi": https://doi.org/10.1007/s00229-009-0270-7. No article title or description of the particular special cases is supplied.
• Supplied literature assessment, checked on 2026-08-21: the question whether arbitrary minimal binomial generators satisfy the uniqueness hypothesis is labeled open beyond the special cases proved by Shibuta and Takagi. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied record assigns statement status corrected_verified and reports that the clean statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and open target. This verification claim is part of the supplied provenance and has not been independently checked here.