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

Let $F\in \mathbb{Z}[x_1,\dots ,x_n]$ be a homogeneous form of degree $d\ge 2$, and $V_F^{\ast }$ the singular locus of the hypersurface $\{\mathbf{x}\in {\mathbb{A}}_{\mathbb{C}}^n:F(\mathbf{x})=0\}$. A longstanding result of Birch states that there is a non-trivial integral solution to the equation $F(\mathbf{x})=0$ provided $n>\dim V_F^{\ast }+(d-1)2^d$ and there is a non-singular solution in $\mathbb{R}$ and ${\mathbb{Q}}_p$ for all primes $p$. In this article, we give a different formulation of this result. More precisely, we replace $\dim V_F^{\ast }$ with a quantity ${\mathcal{H}}_F$ defined in terms of the Hessian matrix of $F$. This quantity satisfies $0\le {\mathcal{H}}_F\le \dim V_F^{\ast }$; therefore, we improve on the aforementioned result of Birch if ${\mathcal{H}}_F<\dim V_F^{\ast }$. We also prove the corresponding result for systems of forms of equal degree.