Birch’s theorem on forms in many variables with a Hessian condition

Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and V∗F the singular locus of the hypersurface {x∈AnC:F(x)=0}. A longstanding result of Birch states that there is a non-trivial integral solution to the equation F(x1,…,xn)=0 provided n>dimV∗F+(d−1)2d, and there is a non-singular solution in R and Qp for all primes p. We give a different formulation of this result. More precisely, we replace dimV∗F with a quantity HF defined in terms of the Hessian matrix of F. This quantity satisfies 0≤HF≤dimV∗F; therefore, we improve on the aforementioned result of Birch if HF<dimV∗F. We also prove the corresponding result for systems of forms of equal degree.