2 problems

Zobin's conjecture. Every ideal $I^m(E)$ in $\mathcal{P}_0^m(\mathbb{R}^n)$ is equal to $I^m(V)$ for some semi-algebraic set $V$ containing the origin.

Lyapunov's conjecture. If the punctual jet of order $s$ of $\Pi$ at $0$ shows that $\Pi$ has no minimum at $0$, then the equilibrium $(0,0)$ is unstable.