  Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a
finite field of characteristic p. For any fixed positive integer s not
divisible by p, we prove that the (normalized) p-adic Newton polygon of the
L-functions of exponential sums of P(x^s) has a tight lower bound which we call
`Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This
Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for
L-function of exponential sum of P(x) and the `Newton polygon' for L-function
of exponential sum of x^s (given by the classical Stickelberger theory). We
prove an analogous Hodge-Stickelberger lower bound for multivariable Laurent
polynomials as well.
  We prove this Hodge-Stickelberger polygon is the limit of generic Newton
polygons of P(x^s) in a sense that was made explicit in the paper.
