Gale duality, decoupling, parameter homotopies, and monodromy

2014 Spring.Numerical Algebraic Geometry (NAG) has recently seen significantly increased application among scientists and mathematicians as a tool that can be used to solve nonlinear systems of equations, particularly polynomial systems. With the many recent advances in the field, we can now routinely solve problems that could not have been solved even 10 years ago. We will give an introduction and overview of numerical algebraic geometry and homotopy continuation methods; discuss heuristics for preconditioning fewnomial systems, as well as provide a hybrid symbolic-numerical algorithm for computing the solutions of these types of polynomials and associated software called galeDuality; describe a software module of bertini named paramotopy that is scientific software specifically designed for large-scale parameter homotopy runs; give two examples that are parametric polynomial systems on which the aforementioned software is used; and finally describe two novel algorithms, decoupling and a heuristic that makes use of monodromy

A general framework for positive solutions to systems of polynomial equations with real exponents

We study positive solutions to parametrized systems of generalized polynomial equations (with real exponents) in $n$ variables and involving $m$ monomials; in compact form, $x \in \mathbb{R}^n_>$ such that $A (c \circ x^B) = 0$ with coefficient matrix $A \in \mathbb{R}^{n' \times m}$, exponent matrix $B \in \mathbb{R}^{n \times m}$, parameter vector $c \in \mathbb{R}^m_>$ (and componentwise product $\circ$). We identify the invariant geometric objects of the problem, namely a polytope $P$ arising from the coefficients and two subspaces representing monomial differences and dependencies. The dimension of the latter subspace (the monomial dependency $d$) is crucial. Indeed, we rewrite the problem in terms of $d$ (monomial) conditions on the coefficient polytope $P$ (which depend on the parameters via $d$ monomials). We obtain the following classification: If $d=0$, solutions exist (for all $c>0$) and can be parametrized explicitly (thereby generalizing monomial parametrizations of ``toric'' solutions). If $d>0$, the treatment of the $d$ conditions on $P$ requires additional objects and methods such as sign-characteristic functions, Descartes' rule of signs for functions, and Wronskians. To demonstrate the relevance of our general framework, we study five examples from reaction network and real fewnomial theory. In particular, we revisit and improve previous upper bounds for the number of positive solutions to (i) $n$ trinomials involving $n+2$ monomials in $n$ variables and (ii) one trinomial and one $t$-nomial in two variables. Our framework allows a unified treatment of multivariate polynomial equations. In fact, it proves useful even in the univariate case. In particular, we provide a solution formula for trinomials, involving discriminants and roots

Constructing polynomial systems with many positive solutions using tropical geometry

The number of positive solutions of a system of two polynomials in two variables defined in the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. In this paper, we prove that a system as above constructed using this method has at most 6 positive solutions. We also show that this bound is sharp. Moreover, using non-transversal intersections of tropical curves, we construct a system as above having 7 positive solutions.Comment: 21 pages, 8 figure