Polyhedral cones and monomial blowing-ups

We show that a polyhedral cone Γ in with apex at 0 can be brought to the first quadrant by a finite sequence of monomial blowing-ups if and only if . The proof is non-trivially derived from the theorem of Farkas–Minkowski. Then, we apply this theorem to show how the Newton diagrams of the roots of any Weierstraß polynomialP(x,z)=zm+h1(x)zm-1+ +hm-1(x)z+hm(x),hi(x) k x1,…,xn [z], are contained in a polyhedral cone of this type.