Newton polygons for certain two variable exponential sums

Abstract

We studies the Newton polygon for the L-function of toric exponential sums attached to a family of two variable generalized hyperkloosterman sum,$f_{t}(x,y)=x^{n}+y+\frac{t}{xy}$ with $t$ the parameter. The explicit Newton polygon is obtained by systematically using Dwork's $\theta_{\infty}$-splitting function with an appropriate choice of basis for cohomology following the method of Adolphson and Sperber[2]. Our result provides a non-trivial explicit Newton polygon for a non-ordinary family of more than one variable with asymptotical behavior, which gives an evidence of Wan's limit conjecture[15].