I was given a positive polynomial function $f(t,x)=\sum_{k=0}^{2n} a_k(x)t^{2n-k}$, where the coefficients $a_j(x)$ are infinitely differentiable funtion of $x\in \mathbb{R}^n$, and $t \geq 0$. My problems are listed below.
- The dimention (or Hausdorff dimention) of the zero set of $f$, denoted by $Z(f):=\{(t,x)\in \mathbb{R}_+ \times \mathbb{R}^n | f(t,x)=0\}$.
Often $Z(f)$ is seen as the union of some continuous curves $t=t(x)$ or some isolated points, but I 'm confused that this set can be rather singular because of $f$ being positive, so that its zero point are not regular point so we cannot apply the "implicit function theorem" to say $Z(f)$ is a smooth submanifold. Can $Z(f)$ be more singular or its sure that the dimention of $Z(f)$ is no greater than 1?
- Find a nice curve to cover the zeroset.
If given my optimal situation that $Z(f)$ is locally described as curves and points, I wish to use numbers of Lipschitz curve to cover $Z(f)$. But I cannot make clear that when isolated points are clustered close together in a singular way (non Lipschitz), can I still find such a nice curve to cover these points? Generally, however, I'm sure that curves in $Z(f)$ can be only Holder continuous rather than Lipschitz.
I appreciate for any help and suggestions!