Explicit generators for saturated ideals in integer-valued polynomial rings

Determine an explicit description of the infinitely many additional generators, in arbitrarily high degrees, arising from the saturation of the ideal generated by the divided-power cocircuit relations in the divided-powers algebra associated with a totally unimodular vector arrangement.

Background

The paper presents the integral internal zonotopal algebra as a quotient of a free divided-powers algebra by a saturated ideal generated by divided-power relations associated with cocircuits. Saturation is required because the initially displayed generators do not generally produce the full integral ideal.

Although the saturated ideal gives the correct integral presentation, the authors note that saturation introduces infinitely many generators in arbitrarily high degrees and explicitly state that they do not know a more explicit description of these generators.

References

Indeed, the operation of saturation involves adding infinitely many new generators in arbitrarily high degrees; we do not know of a more explicit description of these generators.

— The geometry of zonotopal algebras I: cohomology of graphical configuration spaces  (2502.12768 - Crowley et al., 18 Feb 2025) in Section 2, subsection “Presentations,” final remark of the subsection