Determine the higher-order Takens coefficient and exact codimension

Determine the order n of the function g in the Takens normal form, equivalently compute the coefficient of u^2v when b=0, for the isolated degenerate Bogdanov–Takens singularities considered in the cross-product and sparse Kolmogorov families, in order to decide whether their codimension is exactly three rather than merely at least three.

Background

For a nilpotent planar singularity in Takens form u˙=v\dot u=v, v˙=f(u)+vg(u)\dot v=f(u)+v g(u), the paper uses m=ord⁡fm=\operatorname{ord}f and n=ord⁡gn=\operatorname{ord}g to determine the singularity’s codimension. The mixed-volume argument bounds mm, and at the isolated degeneracies with b=0b=0 it establishes m=2m=2, but it provides no information about nn.

Exact codimension three requires n=2n=2, which is equivalent to nonvanishing of the coefficient of u2vu^{2}v in the Takens form. The paper does not compute this coefficient and leaves open the determination of nn, including whether sparse Kolmogorov families can realize prescribed or arbitrarily large values of nn.

References

That coefficient is not computed anywhere in this paper, and by Corollary~\ref{cor:equiv} it cannot be: the mixed volume bounds $m$ and imposes no restriction whatever on $n$. The distinction is immaterial for the topological type, which is a cusp for every $n\ge1$ once $m=2$. We stress what is and is not asserted. The bound leaves $n$ undetermined; it does not exhibit systems in the class realizing arbitrarily large values of $n$. The absence of an upper bound is not an existence statement, and constructing sparse Kolmogorov families with prescribed $n$ is a separate question, not addressed here.

— An intrinsic characterization of the Bogdanov-Takens normal-form coefficients and a mixed-volume obstruction to non-isolated degeneracies  (2608.13931 - Castellanos et al., 14 Aug 2026) in Remark 2.14, Section 3.4 (Remark labeled rem:exactcodim); Conclusions