Polyhedral cones and monomial blowing-ups

We show that a polyhedral cone Γ   in RnRn with apex at 0 can be brought to the first quadrant by a finite sequence of monomial blowing-ups if and only if Γ∩(-R⩾n)={0}. 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),P(x,z)=zm+h1(x)zm-1+⋯+hm-1(x)z+hm(x),hi(x)∈k〚x1,…,xn〛[z]hi(x)∈k〚x1,…,xn〛[z], are contained in a polyhedral cone of this type.