Zur Theorie der quadratischen Formen �ber formalreellen K�rpern

1977

Quadratic Lattices in Function Fields of Genus 0

1993

Beweis des Krullschen Durchschnittsatzes f�r den Wittring

1971

Tsen–Lang Theory for Cpi-fields

1995

Multiplicative Quadratic Forms

1995

An elementary proof of Hilbertʼs theorem on ternary quartics

2012

Abstract In 1888, Hilbert proved that every nonnegative quartic form f = f ( x , y , z ) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f = p 1 2 + p 2 2 + p 3 2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is chosen generically with a real zero. Although these facts were known, there wa…