Polar Cremona Transformations and Monodromy of Polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial $f\in \text{C}[x_0,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ of degree $d$, defined by $${\varphi }_f=grad(f):D(f)\to {\text{CP}}^n,(x_0:\mathrm{.}\mathrm{.}\mathrm{.}:x_n)\to (f_0(x):\mathrm{.}\mathrm{.}\mathrm{.}:f_n(x))$$ where $D(f)={x\in \CP^n; f(x)\neq 0}$istheprincipalopensetassociatedto$f$and$f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of ${\varphi }_f$ under the assumption that the projective hypersurface $V:f=0$ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very strong conditions on the singularities of $V$.