On strong Euler‐homogeneity and Saito‐holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors

Abstract We first develop some criteria for a general divisor to be strongly Euler‐homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito‐holonomicity, generalizing Koszul‐free–type properties and characterizing them in terms of the same Fitting ideals. Thanks to these advances, we are able to make progress in the understanding of a conjecture from 2002: A free divisor satisfying the Logarithmic Comparison Theorem (LCT) must be strongly Euler‐homogeneous. Previously, it was known to be true only for ambient dimension or assuming Koszul‐freeness. We prove it in the following new cases: assuming strong Euler‐homogeneity outside a discrete set of points; assuming the divisor is weakly Koszul‐free; for ; for linear free divisors in . Finally, we refute a conjecture stating that all linear free divisors satisfy LCT, are strongly Euler‐homogeneous and have ‐functions with symmetric roots about .