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**Variations of \(p\)-adic Newton polygons for \(L\)-functions of exponential sums.**
*(English)*
Zbl 1084.11067

Let \(F_q\) be the finite field of \(q\) elements and for \(k\geq 1\) let \(F_{q^k}\) be the extension of \(F_q\) of degree \(k\). For any Laurent polynomial \(f\in F_q[X_1,X_1^{-1},\ldots,X_n,X_n^{-1}]\) form the additive character sum

\[ S_k(f)=\sum_{(x_1,\ldots,x_n)\in (F_{q^k}\setminus \{0\})^n} \psi_k(f(x_1,\ldots,x_n)), \]

where \(\psi_k\) is the additive canonical character of \(F_{q^k}\). With the Laurent polynomial is associated an \(L\)-function

\[ L(f,T)=\exp\left(\sum_{k=1}^\infty S_k(f)\frac{T^k}{k}\right). \]

If \(f\) is not degenerated, this \(L\)-function is a polynomial. A. Adolphson and S. Sperber [Ann. Math. (2) 130, 367–406 (1989; Zbl 0723.14017)] calculated the degree of \(P\) from the Newton polyhedron 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. Combined with his work in [Newton polygons of zeta functions and \(L\) functions. Ann. Math. (2) 137, No. 2, 249–293 (1993; Zbl 0799.11058)] the author proves that this conjecture is true in every dimension \(n\leq 3\) and false for every \(n\geq 4\). Moreover, he shows that a weaker form of the Adolphson-Sperber conjecture is always true. These results are applications of a flexible collapsing decomposition theorem introduced in this paper.

\[ S_k(f)=\sum_{(x_1,\ldots,x_n)\in (F_{q^k}\setminus \{0\})^n} \psi_k(f(x_1,\ldots,x_n)), \]

where \(\psi_k\) is the additive canonical character of \(F_{q^k}\). With the Laurent polynomial is associated an \(L\)-function

\[ L(f,T)=\exp\left(\sum_{k=1}^\infty S_k(f)\frac{T^k}{k}\right). \]

If \(f\) is not degenerated, this \(L\)-function is a polynomial. A. Adolphson and S. Sperber [Ann. Math. (2) 130, 367–406 (1989; Zbl 0723.14017)] calculated the degree of \(P\) from the Newton polyhedron 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. Combined with his work in [Newton polygons of zeta functions and \(L\) functions. Ann. Math. (2) 137, No. 2, 249–293 (1993; Zbl 0799.11058)] the author proves that this conjecture is true in every dimension \(n\leq 3\) and false for every \(n\geq 4\). Moreover, he shows that a weaker form of the Adolphson-Sperber conjecture is always true. These results are applications of a flexible collapsing decomposition theorem introduced in this paper.

Reviewer: Arne Winterhof (Linz)

### MSC:

11T23 | Exponential sums |

11G25 | Varieties over finite and local fields |

11M41 | Other Dirichlet series and zeta functions |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |