Variance of the spectral numbers and Newton polygons

Using the theory of the mixed Hodge structure one can define a notion of spectrum of a singularity or of a polynomial. Recently Claus Hertling proposed a conjecture about the variance of the spectrum of a singularity. Alexandru Dimca proposed a similar conjecture on polynomials. Here, we prove these two conjectures in the case of dimension 2 and when the singularity or the polynomial is Newton non-degenerated and commode.