Logarithmic representatives for rational n-forms

Develop a general, computationally implementable method to construct logarithmic differential-form representatives for arbitrary rational $n$-forms, so that intersection numbers can be evaluated via the logarithmic zero-locus formula in practical applications.

Background

Mizera’s formula for intersection numbers is most effective when cohomology classes have logarithmic representatives, but obtaining such representatives is only straightforward in special cases (e.g., hyperplane arrangements or one variable).

The authors note the lack of a general constructive prescription to convert a given rational form into a logarithmic representative, which limits the broader applicability of the method.

References

However, except in the particular case of a hyperplane arrangement and the trivial case one variable case, there is no known realization of such map, in the sense of a general computationally implementable prescription to find a logarithmic equivalent to a general rational $n-$form.

— Exponential Periods for Integrals in Physics  (2603.29787 - Massidda, 31 Mar 2026) in Subsection "Cohomological intersection numbers"