The nondegenerate-manifold conjecture for rational-point asymptotics

Let $\mathscr{M}$ be an $m$-dimensional manifold in $\mathbb{R}^n$, with co-dimension $c=n-m$, and let $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ be the counting function appearing in the Main Conjecture. A manifold is $l$-nondegenerate at a point when the derivatives through order $l$ of a local parametrization span the ambient space; it is $l$-nondegenerate everywhere when this holds at every point. Nondegeneracy conjecture. If $\mathscr{M}$ is $(m+1)$-nondegenerate everywhere, then

$$\mathrm{N}_{\mathscr{M}}(\delta,Q)\sim c_{\mathscr{M}}\delta^{c}Q^{m+1}\qquad(Q\rightarrow\infty)$$

holds uniformly for

$$\delta\in(Q^{\epsilon-\frac{1}{c}},1/2),$$

for every fixed $\epsilon>0$. This conjecture specifies a natural candidate for the proper curvature conditions in the Main Conjecture. Its resolution is not supplied in the paper.