For any field K with |K| >= 2 and separable L/K of degree binom(n+d, d), there exists P in P^n(L) such that the space of degree-d forms over K vanishing at P has the expected dimension.
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Hypersurfaces passing through the Galois orbit of a point
For any field K with |K| >= 2 and separable L/K of degree binom(n+d, d), there exists P in P^n(L) such that the space of degree-d forms over K vanishing at P has the expected dimension.