Sums of Squares in Algebraic Function Fields

A recently found local-global principle for quadratic forms over function fields of curves over a complete discrete valued field is applied to the study of quadratic forms, sums of squares, and related field invariants.

Choice Reviews Online, 2013

Oberwolfach Reports, 2010

Linear series have long played a central role in algebraic geometry. In recent years, starting with seminal papers by Demailly and Ein-Lazarsfeld, local properties of linear series -in particular local positivity, as measured by Seshadri constants -have come into focus. Interestingly, in their multi-point version they are closely related to the famous Nagata conjecture on plane curves. While a number of important basic results are available by now, there are still a large number of open questions and even completely open lines of research.

2016

Algebraic geometry is one of the central subjects of mathematics. Mathematical physicists, homotopy theorists, complex analysts, symplectic geometers, representation theorists speak the language of algebraic geometry. In this seminar we shall discuss some basic topics of algebraic geometry and their relation with current problems in mathematics. Recommended textbook: J. Harris, Algebraic Geometry: A First Course, I. R. Shafarevich: Basic Algebraic Geometry 1,2, Dolgachev: Introduction to Algebraic Geometry (lecture notes)

kyokan.ms.u-tokyo.ac.jp

2020

Contemporary Mathematics, 2008

We prove that if A ⊂ F q is such that |A| > q 1 2 + 1 2d , then F * q ⊂ dA 2 = A 2 + • • • + A 2 d times, where A 2 = {a • a ′ : a, a ′ ∈ A}, and where F * q denotes the multiplicative group of the finite field F q. In particular, we cover F * q by A 2 + A 2 if |A| > q 3 4. Furthermore, we prove that if |A| ≥ C 1 d size q 1 2 + 1 2(2d−1) , then |dA 2 | ≥ q • C 2 size C 2 size + 1. Thus dA 2 contains a positive proportion of the elements of F q under a considerably weaker size assumption.We use the geometry of F d q , averages over hyper-planes and orthogonality properties of character sums. In particular, we see that using operators that are smoothing on L 2 in the Euclidean setting leads to non-trivial arithmetic consequences in the context of finite fields.

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Mathematische Zeitschrift, 1986

w 1. Introduction Let F be a field of characteristic +2. For any integer n> 1 let gv(n) be the minimum of all r, such that any sum of squares of n-ary F-linear forms is a sum of r squares of n-ary F-linear forms.