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 \langle 1,{a}_{1},\dots ,{a}_{n}\rangle $ is anisotropic
and
$S$ has an
$L$rational point for some
odddegree extension
$L\u2215F$,
then
$S$ has an
$L$rational point for some
odddegree extension
$L\u2215F$
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 \langle x,g\left(x\right)\rangle $ 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
$W\left(F\left(C\right)\u2215F\right)$,
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
$W\left(F\left(C\right)\u2215F\right)$.
