Consider a polynomial in $d$ variables, $p:\mathbb{R}^{d}\rightarrow\mathbb{R}$. Denote by $\mathcal{C}$ its set of zeros, i.e.
$$\mathcal{C}=\{x\in\mathbb{R}^{d}\ |\ p(x)=0\}.$$



> 
> **Q.** Is it possible to find finitely many (not necessarily disjoint) manifolds $M\_{1},\dots, M\_{n}\subset\mathbb{R}^{d}$, with possibly different dimensions, such that
>  $$\mathcal{C}=\bigcup\_{k=1}^{n}M\_{k}?$$
> 
> 
> 


My question arises in the context of degenerate real matrices, namely, the case where $\mathcal{C}$ consists on the set of symmetric real matrices with at least one repeated eigenvalue (in this context, $p(x)$ is the discriminant of the symmetric real matrix $x$). Any suggestion on how to approach this problem or a reference related to the problem will be greatly appreciated.



Zeros of polynomials as a finite union of manifolds
The answer is *yes*. 


Your set $\mathcal{C}$ is by definition a [(semi)algebraic subset](https://en.wikipedia.org/wiki/Semialgebraic_set) of $\mathbb{R}^d$, and any such a subset has a [Whitney stratification](https://en.wikipedia.org/wiki/Whitney_conditions) with finitely many (semi)algebraic strata. 


See A. Dimca, *Singularities and Topology of Hypersurfaces*, [ZBL07535700](https://zbmath.org/?q=an:0753.57001), 
Chapter 1, Corollary 1.12 for more details. 


**Edit.** It is also true that any closed analytic (or subanalytc) set of an analytic manifold admits a Whitney stratification whose strata are analytic submanifolds. In the complex setting, this is called a *complex Whitney stratification*. See 


M. Tib$\check{\rm u}$ar, *Polynomials and vanishing cycles*, Cambridge Tracts in Mathematics **170** (2007), [ZBL1126.32026](https://zbmath.org/?q=an:1126.32026), page 219 and the references given therein.


