Real Enumerative Geometry and Effective Algebraic Equivalence

Determining the common zeroes of a set of polynomials is further complicated over nonalgebraically closed fields such as the real numbers. A special case is whether a problem of enumerative geometry can have all its solutions be real. We call such a problem fully real. Little is known about enumerative geometry from this perspective. A standard proof of Bézout’s Theorem shows the problem of intersecting hypersurfaces in projective space is fully real. Khovanskii [9] considers intersecting hypersurfaces in a torus defined by few monomials and shows the real zeros are at most a fraction of the complex zeroes. Fulton, and more recently, Ronga, Tognoli and Vust [14] have shown the problem of 3264 plane conics tangent to five given conics is fully real. The author [17] has shown all problems of enumerating lines incident on linear subspaces of projective space are fully real. There are few methods for studying this phenomenon. We ask: How can the knowledge that one enumerative problem is fully real be used to infer that a related problem is fully real? We give several procedures to accomplish this inference and examples of their application, lengthening the list of enumerative problems known to be fully real.