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We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and M\"ullerfrom the continuous setting to various discrete settings, obtaining sharpbounds for exponential sums and the number of solutions to polynomialcongruences for general quasi-homogeneous polynomials in two variables. Thisextends work of Denef and Sperber and also Cluckers regarding a conjecture ofIgusa in the two dimensional setting by no longer requiring the polynomial tobe nondegenerate with respect to its Newton diagram.

Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case