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

Abstract

Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and $V_F^*$ the singular locus of the hypersurface $\{\mathbf{x} \in \mathbb{A}^n_{\mathbb{C}}: F(\mathbf{x}) = 0 \}$. A longstanding result of Birch states that there is a non-trivial integral solution to the equation $F(\mathbf{x}) = 0$ provided $n > \dim V_F^* + (d-1) 2^d$ 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 $\dim V_F^*$ with a quantity $\mathcal{H}_F$ defined in terms of the Hessian matrix of $F$. This quantity satisfies $0 \leq \mathcal{H}_F \leq \dim V_F^*$; therefore, we improve on the aforementioned result of Birch if $\mathcal{H}_F < \dim V_F^*$. We also prove the corresponding result for systems of forms of equal degree.