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Register a new userRobbas method on exponential sums
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Peigen Li
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In the present article, we use Robbas method to give an estimation of the Newton polygon for the L function on torus.
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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.
Using $ell$-adic cohomology of tensor inductions of lisse $overline{mathbb Q}_ell$-sheaves, we study a class of incomplete character sums.
Normalized exponential sums are entire functions of the form $$ f(z)=1+H_1e^{w_1z}+cdots+H_ne^{w_nz}, $$ where $H_1,ldots, H_ninC$ and $0<w_1<ldots<w_n$. It is known that the zeros of such functions are in finitely many vertical strips $S$. The asymptotic number of the zeros in the union of all these strips was found by R. E. Langer already in 1931. In 1973, C. J. Moreno proved that there are zeros arbitrarily close to any vertical line in any strip $S$, provided that $1,w_1,ldots,w_n$ are linearly independent over the rational numbers. In this study the asymptotic number of zeros in each individual vertical strip is found by relying on R. J. Backlunds lemma, which was originally used to study the zeros of the Riemann $zeta$-function. As a counterpart to Morenos result, it is shown that almost every vertical line meets at most finitely many small discs around the zeros of $f$.
We give upper bounds for the level and the Pythagoras number of function fields over fraction fields of integral Henselian excellent local rings. In particular, we show that the Pythagoras number of $mathbb{R}((x_1,dots,x_n))$ is $leq 2^{n-1}$, which answers positively a question of Choi, Dai, Lam and Reznick.
In this paper, the formulas of some exponential sums over finite field, related to the Coulters polynomial, are settled based on the Coulters theorems on Weil sums, which may have potential application in the construction of linear codes with few weights.
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