Zariski-openness of the strong-cyclicity locus

Determine whether, beyond analytic openness, the nonvanishing locus of the period determinant characterizing strong cyclicity of a selected rapid-decay contour is Zariski open in the polynomial coefficients.

Background

The paper introduces a period determinant whose nonvanishing is equivalent to a selected rapid-decay cycle being cyclic on a prescribed monodromy-stable subspace of a finite-rank meromorphic connection. On a fixed tame sectorial stratum, holomorphic dependence of the period functions implies that this nonvanishing condition defines an analytic-open locus.

The unresolved issue is whether this analytically open locus is also Zariski open in the coefficients of the polynomial phase. The paper notes that the relevant period functions are generally transcendental, so its argument does not establish an algebraic-openness result.