Polyhedral cones and monomial blowing-ups

Abstract

We show that a polyhedral cone Γ in R n 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 polynomial P ( x , z ) = z m + h 1 ( x ) z m - 1 + ⋯ + h m - 1 ( x ) z + h m ( x ) , h i ( x ) ∈ k 〚 x 1 , … , x n 〛 [ z ] , are contained in a polyhedral cone of this type.