The Bogdanov--Takens normal-form coefficients in $\mathbb{R}^n$ as directional derivatives of the characteristic invariants

Abstract

Let $X$ be a vector field on an open set of $\mathbb{R}^n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the transverse block, we prove that the quadratic coefficients $a,b$ of the Bogdanov--Takens normal form on the centre manifold are $a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$ and $b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}^2}$. The underlying spectral identity requires only invertibility of the transverse block and remains valid without hyperbolicity. Both coefficients arise as the lowest-order terms of a generating identity for the first-order spectral jet of $DX$ along $\ker J$; all higher coefficients depend on the transverse block. We prove that this dichotomy is sharp. The planar formulas $a=-\frac{1}{2}D_{q_0}\det$ and $b=D_{q_0}\operatorname{tr}$ are recovered when $n=2$. We also obtain a coordinate-free nondegeneracy test and a geometric interpretation in terms of the transversality of the kernel line to two invariant hypersurfaces. A self-contained Wolfram Language notebook accompanies the paper and verifies the results symbolically.