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Upper bounds for certain trigonometric sums involving cosine powers

Anastasios D. Simalarides (2015)

Colloquium Mathematicae

We establish upper bounds for certain trigonometric sums involving cosine powers. Part of these results extend previous ones valid for the sum ∑ m = 1 k - 1 | s i n ( π r m / k ) | / s i n ( π m / k ) . We apply our results to estimate character sums in an explicit and elementary way.

Upper bounds for the degrees of decomposable forms of given discriminant

K. Győry (1994)

Acta Arithmetica

1. Introduction. In our paper [5] a sharp upper bound was given for the degree of an arbitrary squarefree binary form F ∈ ℤ[X,Y] in terms of the absolute value of the discriminant of F. Further, all the binary forms were listed for which this bound cannot be improved. This upper estimate has been extended by Evertse and the author [3] to decomposable forms in n ≥ 2 variables. The bound obtained in [3] depends also on n and is best possible only for n = 2. The purpose of the present paper is to establish...

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