  We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the
codimension of a certain variety and adapt the proof to the case of arbitrary
characteristic. We also find a sharper result of this kind for homogeneous
polynomials of degree d (assuming that the characteristic does not divide
d(d-1)). We then use this to relate the Schmidt rank of a homogeneous
polynomial (resp., a collection of homogeneous polynomials of the same degree)
with the codimension of the singular locus of the corresponding hypersurface
(resp., intersection of hypersurfaces). This gives an effective version of
Ananyan-Hochster's Theorem A from arXiv:1610.09268.
