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**Newton polygons of zeta functions and \(L\) functions.**
*(English)*
Zbl 0799.11058

With any Laurent polynomial \(f\) in \(n\) variables over some finite field of characteristic \(p\) is associated an \(L\)-function. If \(f\) is not degenerated, this \(L\)-function is known to be \(P^{(-1)^{n-1}}\) for some polynomial \(P\). A. Adolphson and S. Sperber [Ann. Math., II. Ser. 130, 367-406 (1989; Zbl 0723.14017)] calculated the degree of \(P\) from the Newton polyhedron \(\Delta\) of \(f\) at infinity and gave a lower bound for the Newton polygon of \(P\). Moreover they conjectured that the Newton polygon of the polynomial \(P\) associated with the generic Laurent polynomial \(f\) with given Newton polyhedron at infinity coincides with this lower bound (actually this is conjectured only for \(p\) in some well determined arithmetic progression, for, otherwise, the lower bound gives non integral slopes which is clearly impossible).

This first paper shows that the Adolphson-Sperber conjecture is true for \(\Delta\) if true for each face of \(\Delta\). It also gives explicit computation in the case where \(\Delta\) is an \(n\)-dimensional simplex. The second paper (for bibliographic data see below) shows that trueness of the Adolphson-Sperber conjecture is conserved through more involved decompositions of \(\Delta\) (namely star decomposition and cutting \(\Delta\) by some hyperplane). In this way the general case is reduced to the simplex case. Finally, it is obtained that the Adolphson-Sperber conjecture, although false in its full form (counter-examples are given), is true in a slightly weaker form.

Applications of the method are given for zeta-functions. For instance the, so called, Dwork-Mazur conjecture, saying that the Newton polygon of the “interesting part” of the zeta-function of the generic hypersurface of given degree coincides with its Hodge polygon, is proved.

This first paper shows that the Adolphson-Sperber conjecture is true for \(\Delta\) if true for each face of \(\Delta\). It also gives explicit computation in the case where \(\Delta\) is an \(n\)-dimensional simplex. The second paper (for bibliographic data see below) shows that trueness of the Adolphson-Sperber conjecture is conserved through more involved decompositions of \(\Delta\) (namely star decomposition and cutting \(\Delta\) by some hyperplane). In this way the general case is reduced to the simplex case. Finally, it is obtained that the Adolphson-Sperber conjecture, although false in its full form (counter-examples are given), is true in a slightly weaker form.

Applications of the method are given for zeta-functions. For instance the, so called, Dwork-Mazur conjecture, saying that the Newton polygon of the “interesting part” of the zeta-function of the generic hypersurface of given degree coincides with its Hodge polygon, is proved.

Reviewer: G.Christol (Paris)

### MSC:

11T23 | Exponential sums |

14G15 | Finite ground fields in algebraic geometry |

11G25 | Varieties over finite and local fields |