Vol. 292, No. 1, 2018

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ISSN: 0030-8730
On rational points of certain affine hypersurfaces

Alexander S. Sivatski

Vol. 292 (2018), No. 1, 239–256
Abstract

Let F be a field with charF2, let a1,,an F , and let f F[y] be a monic polynomial of degree 2m. Let further S be an affine hypersurface over F determined by the equation f(y) = i=1naixi2. In the first part of the paper we prove a certain version of Springer’s theorem. Namely, we show that if the form ψ 1,a1,,an is anisotropic and S has an L-rational point for some odd-degree extension LF, then S has an L-rational point for some odd-degree extension LF with [L : F] m, and the last inequality is strict in general.

In the second part we consider the case where the polynomial f is quartic. We show that the surface S has a rational point if and only if the quadratic form ψ x,g(x) is isotropic over F(x), where g(x) F[x] is a certain polynomial of degree at most 3, whose coefficients are expressed in a polynomial way via the coefficients of f.

In the third part we describe all Pfister forms that belong to the Witt kernel W(F(C)F), where C is the plane nonsingular curve determined by the equation y2 = a4x4 + a2x2 + a1x + a0. In the case where the u-invariant of F is at most 10, we describe generators of the ideal W(F(C)F).

Keywords
quadratic form, Springer's theorem, Brauer group, Pfister form, field extension
Mathematical Subject Classification 2010
Primary: 11E04
Secondary: 11E81
Milestones
Received: 7 January 2017
Revised: 13 February 2017
Accepted: 22 March 2017
Published: 22 September 2017
Authors
Alexander S. Sivatski
Departamento de Matemática
Universidade Federal do Rio Grande do Norte
Natal
Brazil