#### Vol. 292, No. 1, 2018

 Recent Issues Vol. 293: 1  2 Vol. 292: 1  2 Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Vol. 288: 1  2 Vol. 287: 1  2 Vol. 286: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear 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 $char\phantom{\rule{0.3em}{0ex}}F\ne 2$, let ${a}_{1},\dots ,{a}_{n}\in {F}^{\ast }$, and let $f\in F\left[y\right]$ be a monic polynomial of degree $2m$. Let further $S$ be an affine hypersurface over $F$ determined by the equation $f\left(y\right)={\sum }_{i=1}^{n}{a}_{i}{x}_{i}^{2}$. In the first part of the paper we prove a certain version of Springer’s theorem. Namely, we show that if the form $\psi \simeq 〈1,-{a}_{1},\dots ,-{a}_{n}〉$ is anisotropic and $S$ has an $L$-rational point for some odd-degree extension $L∕F$, then $S$ has an $L$-rational point for some odd-degree extension $L∕F$ with $\left[L:F\right]\le m$, 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 $\psi \perp 〈-x,g\left(x\right)〉$ is isotropic over $F\left(x\right)$, where $g\left(x\right)\in F\left[x\right]$ 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 $W\left(F\left(C\right)∕F\right)$, where $C$ is the plane nonsingular curve determined by the equation ${y}^{2}={a}_{4}{x}^{4}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$. In the case where the $u$-invariant of $F$ is at most $10$, we describe generators of the ideal $W\left(F\left(C\right)∕F\right)$.

##### Keywords
quadratic form, Springer's theorem, Brauer group, Pfister form, field extension
Primary: 11E04
Secondary: 11E81