A’Campo’s forests for the space of complex polynomials

A new cellulation for the space of complex, polynomials is given. Each polynomial is characterized by A’Campo's ``geometric pictures’’  which are bi-colored planar graphs. These A’Campo forests provide a semi-algebraic stratification for the space. The strata are contractible by Riemann's theorem on the conformal structure of $S^{2}$. Using Łojasiewicz's triangulation, we provide a new cell decomposition. From this cell decomposition follows the cohomology groups for the space of polynomials. This approach is reminiscent of the Grothendieck ``dessin d'enfants'', but is far from the construction of Grothendieck, Penner and  Shabat-Voevodsky, concerning only polynomials having two critical values.