The Newton Polytope of the Implicit Equation

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.Comment: 18 pages, 3 figure

Algorithms for Bohemian Matrices

This thesis develops several algorithms for working with matrices whose entries are multivariate polynomials in a set of parameters. Such parametric linear systems often appear in biology and engineering applications where the parameters represent physical properties of the system. Some computations on parametric matrices, such as the rank and Jordan canonical form, are discontinuous in the parameter values. Understanding where these discontinuities occur provides a greater understanding of the underlying system. Algorithms for computing a complete case discussion of the rank, Zigzag form, and the Jordan canonical form of parametric matrices are presented. These algorithms use the theory of regular chains to provide a unified framework allowing for algebraic or semi-algebraic constraints on the parameters. Corresponding implementations for each algorithm in the Maple computer algebra system are provided. In some applications, all entries may be parameters whose values are limited to finite sets of integers. Such matrices appear in applications such as graph theory where matrix entries are limited to the sets {0, 1}, or {-1, 0, 1}. These types of parametric matrices can be explored using different techniques and exhibit many interesting properties. A family of Bohemian matrices is a set of low to moderate dimension matrices where the entries are independently sampled from a finite set of integers of bounded height. Properties of Bohemian matrices are studied including the distributions of their eigenvalues, symmetries, and integer sequences arising from properties of the families. These sequences provide connections to other areas of mathematics and have been archived in the Characteristic Polynomial Database. A study of two families of structured matrices: upper Hessenberg and upper Hessenberg Toeplitz, and properties of their characteristic polynomials are presented

Numerical Schubert calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

Computation of Real Radical Ideals by Semidefinite Programming and Iterative Methods

Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the $0$ dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but more involved, description of the real radical for systems with real manifolds of solutions (the positive dimensional case) with corresponding advantages in applications. The stable and practical computation of real radicals in the approximate case is an important open problem. Theoretical advances and corresponding implemented algorithms are made for this problem. The approach of the thesis, is to use semidefinite programming (SDP) methods from algebraic geometry, and also techniques originating in the geometry of differential equations. The problem of finding the real radical is re-formulated as solving an SDP problem. This approach in the $0$ dimensional case, was pioneered by Curto \& Fialkow with breakthroughs in the $0$ dimensional case by Lasserre and collaborators. In the positive dimensional case, important contributions have been made of Ma, Wang and Zhi. The real radical corresponds to a generic point lying on the intersection of boundary of the convex cone of semidefinite matrices and a linear affine space associated with the polynomial system. As posed, this problem is not stable, since an arbitrarily small perturbation takes the point to an infeasible one outside the cone. A contribution of the thesis, is to show how to apply facial reduction pioneered by Borwein and Wolkowicz, to this problem. It is regularized by mapping the point to one which is strictly on the interior of another convex region, the minimal face of the cone. Then a strictly feasible point on the minimal face can be computed by accurate iterative methods such as the Douglas-Rachford method. Such a point corresponds to a generic point (max rank solution) of the SDP feasible problem. The regularization is done by solving the auxiliary problem which can be done again by iterative methods. This process is proved to be stable under some assumptions in this thesis as the max rank doesn\u27t change under sufficiently small perturbations. This well-posedness is also reflected in our examples, which are executed much more accurately than by methods based on interior point approaches. For a given polynomial system, and an integer $d \u3e 0$, Results of Curto \& Fialkow and Lasserre are generalized to give an algorithm for computing the real radical up to degree $d$. Using this truncated real radical as input to critical point methods, can lead in many cases to validation of the real radical