An intrinsic characterization of the Bogdanov-Takens normal-form coefficients and a mixed-volume obstruction to non-isolated degeneracies

Abstract

Let $X$ be a planar vector field with an equilibrium $p$ at which the Jacobian $J=DX(p)$ is nilpotent of rank one, and let $q_0$ span $\ker J$. We prove that the two coefficients $a$ and $b$ of the Bogdanov--Takens (BT) normal form are the directional derivatives, along $q_0$, of the two invariants of the Jacobian: $a=-\frac{1}{2}\langle\nabla\det DX(p),q_0\rangle$, $b=\langle\nabla\operatorname{tr}DX(p),q_0\rangle$. The identity is invariant under changes of phase-space coordinates and equivariant under the rescaling of $q_0$, and the resulting formula requires neither generalized eigenvectors nor the second-order multilinear form. It yields a coordinate-free reading of the BT nondegeneracy conditions in terms of the kernel line field of the projection of the equilibrium manifold onto parameter space, the transformation rule $(a,b)\mapsto(h^2a,hb)$ under orbital equivalence, and the fact that $a=0$ whenever the vector field factors through a function vanishing at $p$. We then prove an obstruction of a combinatorial nature. For a Kolmogorov system $\dot{x}=xA/g_1$, $\dot{y}=yB/g_2$ and an equilibrium $p$ in the torus $(\mathbb{C}^*)^2$ at which the Jacobian is nilpotent and nonzero, the order $m=\operatorname{ord}f$ in the Takens normal form is bounded by the mixed volume of the Newton polytopes of $A$ and $B$. In particular, if $\operatorname{MV}(\operatorname{Newt}A,\operatorname{Newt}B)\leq2$ then $a\neq0$ unless the equilibrium fails to be isolated, and the nilpotent singularities of saddle, focus and elliptic type are unreachable: only cusp-type singularities occur at isolated equilibria, while their exact codimension is not controlled by the mixed-volume bound. The class of systems with cross-product cubic terms, for which the mixed volume equals $2$, is treated in detail, and two classical Bazykin models are shown to be instances.