RATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS

Let k = Fq(t) be the rational function field over Fq and f(x) ∈ k[x1, . . . , xs] be a form of degree d. For l ∈ N, we establish that whenever s > l+ d ∑ w=1 w ( d− w + l − 1 l − 1 ) , the projective hypersurface f(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 a of f(x) there are infinitely many s-tuples ($1, . . . , $s) of monic irreducible polynomials over k, with the $i not all equal, and f(a1$1, . . . , as$s) = 0. We establish in fact more general results of the above type for systems of forms over Ci-fields.