Todd Cochrane

On a trigonometric inequality of vinogradov

The sum f(m, n)=∑a=1m−1(|sinπanm||sinπam|) arises in bounding incomplete exponential sums. In this article we show that for positive integers m, n with m>1, f(m, n)<(4π2) m log m+0.38m+0.608+0.116 d2m, where d=(m, n). This improves earlier bounds for f(m, n). The constant 4π2 in the main term is shown to be best possible.

Upper bounds on a two-term exponential sum*

AbstractWe obtain upper bounds for mixed exponential sums of the type

BOUNDS ON EXPONENTIAL SUMS AND THE POLYNOMIAL WARING PROBLEM MOD p

S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)

On upper bounds of Chalk and Hua for exponential sums

where pm is a prime power with m⩾ 2 and X is a multiplicative character (mod pm). If X is primitive or p⫮(a, b) then we obtain |S(χ,f,pm)| ⩽2np2/3 m. If X is of conductor p and p⫮( a, b) then we get the stronger bound |S(χ,f,pm)|⩽npm/2.

Small zeros of quadratic forms modulo p

For a sparse polynomial f(x) = Σ r i=1 a i x k i ∈ Z[x], with p a i and 1 ≤ k 1 < ... < k r < p - 1, we show that formula math thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.

Decimations of

TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE Abstract. We give estimates for the exponential sum ∑p x=1 exp(2πif(x)/p), p a prime and f a non-zero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod p; namely the smallest γ such that f(x1) + · · ·+ f(xγ) ≡ N (mod p) is solvable in integers for any N .

\ell

Let Q(x) = Q(x 1 x 2,…, x n) be a quadratic form with integer coefficients and p be an odd prime. Let µ=µ(Q,p) be minimal such that there is a nonzero x∈Z n with max |x i|≤µ and