Newton Polygons and Topology of Real Zero Loci of Real Polynomials

Let f ( x , y )=∑ a ij x i y 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 )=∑ ( i , j )∈γ a ij x i y 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.