A note on L-series and Hodge spectrum of polynomials

We compare on the one hand the combinatorial procedure described in [1] which gives a lower bound for the Newton polygon of the $L$-function attached to a commode, non-degenerate polynomial with coefficients in a finite field and on the other hand the procedure which gives the Hodge theoretical spectrum at infinity of a polynomial with complex coefficients and with the same Newton polyhedron. The outcome is that they are essentially the same, thus providing a Hodge theoretical interpretation of the Adolphson-Sperber lower bound which was conjectured in [1].