On character sums of rational functions over local fields

Index bounds for character sums of polynomials over finite fields

Abstract. We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings of small indices. As an application, we also give some general bounds for numbers of solutions of some Artin-Schreier equations and mininum weight...

Bounding the Rational Sums of Squares over Totally Real Fields

Let K be a totally real Galois number field. C. J. Hillar proved that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of N(m) squares in Q[x1, . . . , xn], where N(m) ≤ 2[K:Q]+1 · `[K:Q]+1 2 ́ ·4m, the proof being constructive. We show in fact that N(m) ≤ (4[K : Q]−3)·m, the proof being constructive as well.

Character Sums in Finite Fields

Let F q be a finite field of order q with q = p n , where p is a prime. A multiplicative character χ is a homomorphism from the multiplicative group F * q , ·· to the unit circle. In this note we will mostly give a survey of work on bounds for the character sum x χ(x) over a subset of F q. In Section 5 we give a nontrivial estimate of character sums over subspaces of finite fields. §1. Burgess'...

On Certain Character Sums over Smooth Numbers

Sums of Squares over Totally Real Fields Are Rational Sums of Squares

Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of 4m · 2[L:Q]+1 ([L : Q] + 1 2 ) squares in Q[x1, . . . , xn]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebra...