Two improvements in Brauer's theorem on forms

Let $k$ be a Brauer field, that is, a field over which every diagonal form insufficiently many variables has a nonzero solution; for instance, $k$ could bean imaginary quadratic number field. Brauer proved that if $f_1, \ldots, f_r$are homogeneous polynomials on a $k$-vector space $V$ of degrees $d_1, \ldots,d_r$, then the variety $Z$ defined by the $f_i$'s has a non-trivial $k$-point,provided that $\dim{V}$ is sufficiently large compared to the $d_i$'s and $k$.We offer two improvements to this theorem, assuming $k$ is infinite. First, weshow that the Zariski closure of the set $Z(k)$ of $k$-points has codimension$

Language(s)English

Seeing content that should not be on Zendy? Contact us.