Testing the logarithmic comparison theorem for free divisors

We propose in this work a computational criterion to test if a free divisor D subset of C-n verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of D in C-n. For Spencer free divisors D equivalent to (f = 0), we solve a conjecture about the generators of the annihilating ideal of 1/f and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.