A note on the discriminant

Let $F(X,Y)=Y^d+a_1(X)Y^{d-1}+\mathrm{.}\mathrm{.}\mathrm{.}+a_d(X)$ be a polynomial in $n+1$ variables $(X,Y)=(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n,Y)$ with coefficients in an algebraically closed field K. Assuming that the discriminant $D(X)={\text{disc}}_{Y}F(X,Y)$ is nonzero we investigate the order ${\text{ord}}_{P}D$ for $P\in K^n$. As application we get a discriminant criterion for the hypersurface F=0 to be nonsingular.