Two improvements in Birch's theorem on forms

Let K be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, K could be the field of rational numbers. Let f₁, , fᵣ be homogeneous forms of odd degree over K in n variables, and let Z be the variety they cut out. Birch proved if n is sufficiently large then Z (K) contains a non-zero point. We prove two results which show that Z (K) is actually quite large. First, the Zariski closure of Z (K) has bounded codimension in Aⁿ. And second, if the fᵢ's have sufficiently high strength then Z (K) is in fact Zariski dense in Z. The proofs use recent results on strength, and our methods build on recent work of Bik, Draisma, and Snowden, which established similar improvements to Brauer's theorem on forms.