Rocky Mountain Journal of Mathematics
Let k = F-q(t) be the rational function fi eld over F-q and f(x) is an element of k[x(1),..., x(s)] be a form of degree d. For l is an element of N, we establish that whenever s > l + Sigma(d)(w=1)w(2)(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 in fi nitely many s-tuples (pi(1),...,pi(s) ) of monic irreducible polynomials over k, with the pi(i) not all equal, and f(a(1)pi(1),..., a(s)pi(s)) = 0. We establish in fact more general results of the above type for systems of forms over C-i-fields.
Cochrane, Todd; Spencer, Craig V.; and Yang, Hee-Sung, "Rational Linear Spaces on Hypersurfaces over Quasi-Algebraically Closed Fields" (2014). Open Dartmouth: Faculty Open Access Articles. 2468.