NEWTON POLYGONS AND TOPOLOGY OF REAL ZERO LOCI OF REAL POLYNOMIALS

Abstract

Let f(x, y) =[sum ]a_ijxⁱy^j be a real polynomial, and let Δ(f) be the Newton polygon of f, that is, the convex hull of the set of points (i, j) with a_ij≠0. In this paper, we study the relation between the topology of the real zero locus of f and the Newton polygon Δ(f) of f.

Obviously, the Newton polygon Δ(f) is an integral convex polygon. Here, an integral polygon is a polygon with vertices that are integral points. A polynomial f(x, y) is said to be non-degenerate if the gradient of f_γ(x, y) =[sum ]_{(i, j)∈γ}a_ijxⁱy^j has no zeros in (C−0)² for each face γ of Δ(f). If f is non-degenerate, then the zero locus of f can be compactified in a suitable toric surface P_Δ(K) (K=R, C) as a non-singular algebraic curve, and we denote the compactifications by Z(R), Z(C). Here, toric surfaces form a class of algebraic surfaces which contains the affine plane, the projective plane, the product of two projective lines, and so on. In Section 1, we give a review of toric varieties.