  We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and M\"uller
from the continuous setting to various discrete settings, obtaining sharp
bounds for exponential sums and the number of solutions to polynomial
congruences for general quasi-homogeneous polynomials in two variables. This
extends work of Denef and Sperber and also Cluckers regarding a conjecture of
Igusa in the two dimensional setting by no longer requiring the polynomial to
be nondegenerate with respect to its Newton diagram.
