Rational linear spaces on hypersurfaces over quasi-algebraically closed fields

Abstract

Let $k=\mathbb{F}_q(t)$ be the rational function field over $\mathbb{F}_q$ and $f(\mathbf{x}) \in k[x_1, \ldots, x_s]$ be a form of degree~$d$. For $l \in \mathbb{N}$, we establish that whenever \[ s > l + \sum_{w=1}^{d} w^2 \binom{d-w+l-1}{l-1}, \] the projective hypersurface $f(\mathbf{x})=0$ contains a $k$-rational linear space of projective dimension~$l$. We also show that if $s> 1+ d(d+1)(2d+1)/6$, then for any $k$-rational zero $\mathbf{a}$ of $f(\mathbf{x})$ there are infinitely many $s$-tuples $(\varpi_1, \ldots, \varpi_s)$ of monic irreducible polynomials over $k$, with the $\varpi_i$ not all equal, and $f(a_1\varpi_1, \ldots, a_s \varpi_s) =0$. We establish in fact more general results of the above type for systems of forms over $C_i$-fields.