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Let $p$ be a prime number. Every two-variable polynomial $f(x_1, x_2)$ over afinite field of characteristic $p$ defines an Artin--Schreier--Witt tower ofsurfaces whose Galois group is isomorphic to $\mathbb Z_p$. Our goal of thispaper is to study the Newton polygon of the $L$-functions associated to afinite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hullis a fixed triangle $\Delta$. We denote this polygon by $\textrm{GNP}(\Delta)$.We prove a lower bound of $\textrm{GNP}(\Delta)$, which we call the improvedHodge polygon $\textrm{IHP}(\Delta)$, and we conjecture that$\textrm{GNP}(\Delta)$ and $\textrm{IHP}(\Delta)$ are the same. We show that if$\textrm{GNP}(\Delta)$ and $\textrm{IHP}(\Delta)$ coincide at a certain point,then they coincide at infinitely many points.  When $\Delta$ is an isosceles right triangle with vertices $(0,0)$, $(0, d)$and $(d, 0)$ such that $d$ is not divisible by $p$ and that the residue of $p$modulo $d$ is small relative to $d$, we prove that $\textrm{GNP}(\Delta)$ and$\textrm{IHP}(\Delta)$ coincide at infinitely many points. As a corollary, wededuce that the slopes of $\textrm{GNP}(\Delta)$ roughly form an arithmeticprogression with increasing multiplicities.

Generic Newton polygon for exponential sums in two variables with triangular base