The classic Milnor-Thom upper bound on the sum of the Betti numbers of real algebraic sets (for a nice exposition and references, see e.g. N.R. Wallach, *On a Theorem of Milnor and Thom*, in S. Gindikin (ed.), *Topics in Geometry: In Memory of Joseph D'Atri*, Progress in Nonlinear Differential Equations and their Applications **20** (Birkhäuser, 1996), pp. 331-348) allows one to give an upper bound on the number of connected components of the complement $$U_P=\{\xi\in\mathbb{R}^n\ |\ P(\xi)\neq 0\}$$ of the zero set of a real polynomial function $P:\mathbb{R}^n\rightarrow\mathbb{R}$, since $U$ is homeomorphic to the algebraic subset $\tilde{U}_P=\{(\xi,\lambda)\in\mathbb{R}^{n+1}\ |\ \lambda P(\xi)=1\}$ of $\mathbb{R}^{n+1}$ through the map $$U_P\ni\xi\mapsto(\xi,P(\xi)^{-1})\in\tilde{U}_P\ .$$ The aforementioned bound tells us that if the degree of $P$ is $r$, then the sum of the Betti numbers of $U_P$ (and, therefore, its number of connected components) is bounded above by $(r+1)(2r+1)^n$.

> **Question:** is there an upper bound on the number of *convex* connected components of $U_P$ which is sharper than the Milnor-Thom bound but also only depends on $n$ and $r$?

My intuition is that the Milnor-Thom bound is too crude to this end, even in the exceptional case when all connected components of $U_P$ are convex - take, for instance, $P(\xi)=\prod^n_{j=1}\xi_j$, in which case $r=n$ but the connected components of $U_P$ are precisely the $2^n$ orthants $$\mathbb{R}^n_I\doteq\{\xi\in\mathbb{R}^n\ |\ \xi_j>0\text{ if }j\in I\ ,\,\xi_j<0\text{ if }j\not\in I\}\ ,\quad I\subset\{1,\ldots,n\}$$ of $\mathbb{R}^n$, which are clearly convex.

My motivation for the above question is related to Lars Garding's theory of hyperbolic polynomials (see e.g. L. Garding, *An Inequality for Hyperbolic Polynomials*, J. Math. Mech. **8** (1959) 957-965). Recall that an homogeneous polynomial $P:\mathbb{R}^n\rightarrow\mathbb{R}$ is said to be *hyperbolic* with respect to $0\neq\tau\in\mathbb{R}^n$ if $P(\tau)\neq 0$ and the roots of the one-variable polynomial $P_{\xi,\tau}(\lambda)\doteq P(\xi-\lambda\tau)$ are all real. The *hyperbolicity cone* $C(P,\tau)$ to which $\tau$ belongs is the connected component of $U_P$ to which $\tau$ belongs. Garding has shown that $$C(P,\tau)=\{\xi\in\mathbb{R}^n\ |\ \text{all roots of }P_{\xi,\tau}\text{ are positive}\}$$ and, as a consequence, that $C(P,\tau)$ is an open convex cone. Roughly speaking, dual cones of hyperbolicity cones provide upper bounds to the propagation speed of the support of solutions of hyperbolic partial differential equations with principal symbol $i^rP$.

Due to the homogeneity of $P$, hyperbolicity cones always come in opposite pairs - more precisely, if $P$ is hyperbolic with respect to $\tau$, then it also is with respect to $-\tau$, and $C(P,-\tau)=-C(P,\tau)$. Since $P$ in the above example is hyperbolic with respect to any $\tau$ in $U_P$, we conclude that the number of hyperbolicity cones of an hyperbolic homogeneous polynomial $P:\mathbb{R}^n\rightarrow\mathbb{R}$ is an even number bounded above by the number of convex connected components of $U_P$, and such an upper bound is sharp. Therefore, a positive answer to the above question would provide an upper bound to the number of hyperbolicity cones of $P$ which is sharper than the Milnor-Thom bound.