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].


Introduction
Let p be a prime number, put q = p a . For any i 1, denote by F q i the finite field with q i elements (for i = 1 we write F q = k). Let tr i : F q i −→ k denote the trace map. To a polynomial f ∈ k[x 1 , . . . , x n ] and a non-trivial additive character ψ : k −→ C * one attaches the sequence of exponential sums S i (f, ψ) = x∈(F q i ) n ψ tr i (f (x)) (i 1) .

The generating function
is an Artin L-series which encodes much of the information about the arithmetic properties of the polynomial f (see e.g. [6]). If f is commode and non-degenerate (precise definitions are recalled below), it was proven in [1] and [2] that L(f, ψ) (−1) n+1 is a polynomial in one variable whose roots are algebraic over Q. It was also shown in [1] that the Newton polygon of L(f, ψ) (−1) n+1 is bounded below by a polygonal line that we denote LB(f ) and which can be descibed in a purely combinatorial way from the Newton polyhedron of f . This result gives non-trivial information about the p-adic absolute values of the roots of L(f, ψ) (−1) n+1 .
On the other hand, if f is a polynomial with complex coefficients, one can attach to it a variation of mixed Hodge structures and also a limit mixed Hodge structure at infinity. Similar to the case of germs of holomorphic functions, the action of the semi-simple part of the monodromy (about infinity) on the limit Hodge filtration gives rise to a set of rational numbers Spec H (f ), called the spectrum at infinity of the polynomial f . If the polynomial is commode and non-degenerate, C. Sabbah proved in [10], [11] that the spectrum at infinity can also be computed in a combinatorial way.
Let f ∈ Z[x 1 , . . . , x n ] be a commode, non-degenerate polynomial. Then, for all but a finite number of prime numbers p, the reduction f of f modulo p will also be commode and non-degenerate, with the same Newton polyhedron as f . In this note we compare the combinatorial formulas giving the Hodge spectrum of f and those which give the lower bound of the Newton polygon of L(f , ψ) (−1) n+1 . The outcome is that the datum of Spec H (f ) is equivalent to that of the lower bound LB(f ), what provides a Hodge-theoretical interpretation of the later. This interpretation was conjectured in the introduction to [1].
A natural question is whether the relation between the Hodge spectrum of f and the p-adic absolute values of the roots of L(f , ψ) (−1) n+1 is a special feature of the commode, non-degenerate case, or if there is a deeper relation, holding in greater generality. We have no results about this matter, but at the end of this note we present some examples against which the question might be tested. In a different setting, connections between the p-adic valuations of the eigenvalues of Frobenius acting in De Rham cohomology and Hodge theory appear already in [8].
The theorems from [1] and [10] quoted in this note are not stated in their most general form. In particular, some of the theorems we recall which concern commode and non-degenerate polynomials hold also for Laurent polynomials, under suitable hypotheses. For the convenience of the reader, we have tried to keep the same notation as in [1] and [3].

L-series attached to CND polynomials
The class of commode polynomials, non-degenerate with respect to their Newton boundary, was defined by A. Kouchnirenko in [7]. In order to recall his definition, we introduce some notation: Let k be any field, let k denote an algebraic closure of k, let f ∈ k[x 1 , . . . , x n ] be a polynomial. Write The polynomial f is said to be commode (or convenient in some articles) if ∆(f ) intersects all coordinate axes in points different from (0, . . . , 0). If f is simultaneously commode and non-degenerate we will say that f is a CND polynomial.
Let k be a finite field, f ∈ k[x 1 , . . . , x n ] a CND polynomial, ψ : k −→ C * a non-trivial additive character. The following is proved in [1] and [2]: i) L(f, ψ) (−1) n+1 is a polynomial and its roots are algebraic over Q. ii) For any embedding Q ֒→ C, the complex absolute value of the roots of iii) The roots of L(f, ψ) (−1) n+1 are ℓ-adic units for any ℓ = p and any embedding Q ֒→ Q ℓ . The results of Adolphson and Sperber in [1] give information about the possible p-adic absolute values of the roots of L(f, ψ) (−1) n+1 . Their theorem is stated in terms of Newton polygons for polynomials in one variable (these Newton polygons have nothing to do with the Newton polyhedra introduced before, but this is the standard terminology and we will also use it). We recall the definition: Let Ω p be the completion of an algebraic closure of Q p . Denote by "ord" the additive valuation of Ω p normalized by the condition ord(q) = 1. Given h(t) = 1 + a 1 t + . . . a k t k ∈ Ω p [t], the Newton polygon of h(t), denoted N (h(t)), is the lower boundary of the convex envelope in R 2 of the set It is well-known that N (h(t)) has a side of slope s and horizontal projection of length l if and only if h(t) has exactly l roots α ∈ Ω p (counting multiplicities) with ord(α) = −s. Thus, the knowledge of N (p(t)) is equivalent to that of the p-adic valuations of the roots of p(t).
2.1. Assume now that k is an arbitrary field and f ∈ k[x 1 , . . . , x n ] is a CND polynomial, put R = k[x 1 , . . . , x n ]. For any face σ of ∆(f ) denote by L σ the unique linear form with coefficients in Q such that L σ ≡ 1 on σ. Given g ∈ R, put w σ (g) := max J L σ (J) where J runs over the multiindexes corresponding to exponents of monomials appearing in g with a non-zero coefficient. Consider the weight function w(g) = max σ w σ (g). Let M be an integer such that w(R) ⊂ M −1 ·N. Define an increasing filtration on R by: Theorem. If k is a finite field and f ∈ k[x 1 , . . . , x n ] is a CND polynomial, then the Newton polygon of L(f, ψ) (−1) n+1 lies on or above the Newton polygon of the polynomial nM i=0 (1 − q i/M t) bi . Remarks.
i) In [1] it is proven also that the endpoints of both Newton polygons coincide. ii) In [1] the filtration considered above is defined (for more general polynomials than CND's) on a certain subring R(f ) of R. For commode polynomials,

Complex polynomials
Let f ∈ C[x 1 , . . . , x n ] be a polynomial. There exists a finite subset Σ ⊂ C such that the restriction of f is a locally trivial smooth fibration (see e.g. [9]. This is not straightforward due to the fact that f is not proper), and the smallest set Σ f verifying this condition is called the set of bifurcation values of f . It follows that we can speak of the topological type of the generic fiber of f , and it can be proven that for polynomials which are CND, this generic fiber has the homotopy type of a bouquet of (n − 1)dimensional spheres ([7, Théorème V]).
The cohomology spaces H n−1 (f −1 (t), Q) carry mixed Hodge structures and the fibration considered above defines a variation of mixed Hodge structures over C−Σ f . In this situation, one can define a limit mixed Hodge structure at infinity (by the results in [13], cf. [5]) and this allows one to define a spectrum, the so-called spectrum at infinity. More precisely, let F • be the limit Hodge filtration and T ∞ ss the semisimple part of the monodromy at infinity. The Hodge spectrum is the finite set of rational numbers with multiplicities, denoted Spec H (f ), defined by the following condition: The number of times that a ∈ Q appears in the spectrum is the dimension of the space of eigenvectors of T ∞ ss of eigenvalue exp(−2πia), where T ∞ ss is regarded as an endomorphism of the quotient F [n−a+1] /F [n−a+1]+1 and where [ · ] denotes integer part (cf. [12, (2.1)] in the local case, notice that we have introduced a shifting of +1 with respect to Steenbrink's definition).
3.1. In [10], [11] C. Sabbah introduced the class of cohomologically tame polynomials (a large class which includes the CND polynomials). For these polynomials, he gives a D-module theoretic interpretation of the Hodge spectrum, and he proves that if f is a CND polynomial, the spectrum can be computed from the Newton polyedron of f . The combinatorial procedure is described in [3] (we will use slightly different notations as those in [3]): Put R = C[x 1 , . . . , x n ]. Let w be the same weight function in R defined in section 2, define w * : R → Z by w * (g) := w(x 1 . . . x n g). For α ∈ Q, put is a graded algebra and the Hodge spectrum of f is given by the following rule: The number of times α ∈ Q appears in Spec H (f ) is the dimension of the α-graded piece of the graded ring (In [3] the weight function used is −ω, which makes the spectral numbers have the opposite sign of those we consider). Let R S , I S be defined as in section 2, taking as field of coefficients the field C. Multiplication by the monomial x 1 . . . x n induces an isomorphism of graded vector spaces where the graded ring on the right hand side was defined in section 2 (see [3, pg. 321, Lemme]. On the other hand, it is a standard fact that as graded rings, where in R/J(f ) one considers the graduation attached to the filtration induced by V • R.
Proposition. Let f ∈ Z[x 1 , . . . , x n ] be a polynomial. Denote f the reduction of f modulo p and assume that both f ∈ F p [x 1 , . . . , x n ] and f ∈ C[x 1 , . . . , x n ] are CND polynomials with the same Newton polyedron. Then Spec H (f ) = Spec AS (f ).
In particular, the polygonal line LB(f ) is determined by the Hodge spectrum of f .
Proof. By the theorem of Sabbah recalled in (3.1), the multiplicity of α ∈ Q in By (1) and isomorphisms (2), (3), this multiplicity coincides with the dimension of the α-th graded piece of the graded ring R S /I S , defined as in (2.1) with k = C. As noted in (2.1), this dimension is independent of the coefficient field.
Since f and f are CND polynomials with the same Newton polyedron, it coincides with the multiplicity of α in Spec AS (f ) (See definition 3). It follows that Spec H (f ) = Spec AS (f ). The last statement follows from the equality of spectra and the definition of the lower bound LB(f ) in remark iii) above.