Wan, Daqing Variations of \(p\)-adic Newton polygons for \(L\)-functions of exponential sums. (English) Zbl 1084.11067 Asian J. Math. 8, No. 3, 427-471 (2004). 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. Reviewer: Arne Winterhof (Linz) Cited in 1 ReviewCited in 43 Documents 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) Keywords:Laurent polynomial; Newton polyhedron; Newton polygon; Adolphson-Sperber conjecture; decomposition theorem; L-function; zeta functions, exponential sums Citations:Zbl 0723.14017; Zbl 0799.11058 × Cite Format Result Cite Review PDF Full Text: DOI