Real solutions to systems of polynomial equations and parameter continuation

Given a parameterized family of polynomial equations, a fundamental question is to determine upper and lower bounds on the number of real solutions a member of this family can have and, if possible, compute where the bounds are sharp. A computational approach to this problem was developed by Dietmaier in 1998 who used a local linearization procedure to move in the parameter space to change the number of real solutions. He used this approach to show that there exists a Stewart-Gough platform that attains the maximum of forty real assembly modes. Due to the necessary ill-conditioning near the discriminant locus, we propose replacing the local linearization near the discriminant locus with a homotopy-based method derived from the method of gradient descent arising in optimization. This new hybrid approach is then used to develop a new result in real enumerative geometry.

Discrete Wilson Lines in F-Theory

F-theory models are constructed where the7-brane has a nontrivial fundamental group. The base manifolds used are a toric Fano variety and a smooth toric threefold coming from a reflexive polyhedron. The discriminant locus of the elliptically fibered Calabi-Yau fourfold can be chosen such that one irreducible component is not simply connected (namely, an Enriques surface) and supports a non-Abelian gauge theory.

GEOMETRY OF CRITICAL LOCI

Let(formula here)be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.For each α ∈ A, we define a rational number dα by(formula here)where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.