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We deduce Katzs theorems for $(A,B)$-exponential sums over finite fields using $ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson-Sperbers bound for the degree of $L$-functions. Applying the facial decomposition theorem in cite{W1}, we prove that the universal family of $(A,B)$-polynomials is generically ordinary for its $L$-function when $p$ is in certain arithmetic progression.
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In this paper, we consider the following $(A, B)$-polynomial $f$ over finite field: $$f(x_0,x_1,cdots,x_n)=x_0^Ah(x_1,cdots,x_n)+g(x_1,cdots,x_n)+P_B(1/x_0),$$ where $h$ is a Deligne polynomial of degree $d$, $g$ is an arbitrary polynomial of degree $< dB/(A+B)$ and $P_B(y)$ is a one-variable polynomial of degree $le B$. Let $Delta$ be the Newton polyhedron of $f$ at infinity. We show that $Delta$ is generically ordinary if $pequiv 1 mod D$, where $D$ is a constant only determined by $Delta$. In other words, we prove that the Adolphson--Sperber conjecture is true for $Delta$.
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