Numerical Algebraic Geometry Research Articles

Combinatorial excess intersection

We provide formulas and develop algorithms for computing the excess numbers of an ideal. The solution for monomial ideals is given by the mixed volumes of polytopes. These results enable us to design numerical algebraic geometry homotopies to compute excess numbers of any ideal.

Perturbed homotopies for finding all isolated solutions of polynomial systems

Given a polynomial system f:CN→Cn, the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f(z)=0, as well as points on any positive-dimensional components of the solution set, V(f). Some of these methods are guaranteed to find all isolated solutions (nonsingular and singular alike), while others may not find singular solutions. One of the most recent advances in this field is regeneration, an equation-by-equation solver that is often more efficient than other methods. However, the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation.The aim of this article is two-fold. First, more generally, we consider the use of perturbed homotopies for solving polynomial systems. In particular, we propose first solving a perturbed version of the polynomial system, followed by a parameter homotopy to remove the perturbation. Such perturbed homotopies are sometimes more efficient than regular homotopies, though they can also be less efficient. Second, a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions, including all singular isolated solutions. This version of regeneration – perturbed regeneration – can decrease the efficiency of regeneration but increases its applicability.

Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. Westudy this problem on manifolds of matrices with bounded rank. These represent mixtures of distributionsof two independent discrete random variables. We determine the maximum likelihood degree for a rangeof determinantal varieties, and we apply numerical algebraic geometry to compute all critical points oftheir likelihood functions. This led to the discovery of maximum likelihood duality between matrices ofcomplementary ranks, a result proved subsequently by Draisma and Rodriguez.

The ideal of the trifocal variety

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.

Numerical elimination and moduli space of vacua

We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm to extract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry.

Isosingular Sets and Deflation

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom---Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.

Cell Cycle Control and Bifurcation for a Free Boundary Problem Modeling Tissue Growth

We consider a free boundary problem for a system of partial differential equations, which arise in a model of cell cycle with a free boundary. For the quasi steady state system, it depends on a positive parameter $$\beta $$ , which describes the signals from the microenvironment. Upon discretizing this model, we obtain a family of polynomial systems parameterized by $$\beta $$ . We numerically find that there exists a radially-symmetric stationary solution with boundary $$r = R$$ for any given positive number $$R$$ by using numerical algebraic geometry method. By homotopy tracking with respect to the parameter $$\beta $$ , there exist branches of symmetry-breaking stationary solutions. Moreover, we proposed a numerical algorithm based on Crandall---Rabinowitz theorem to numerically verify the bifurcation points. By continuously changing $$\beta $$ using a homotopy, we are able to compute non-radially symmetric solutions. We additionally discuss control function $$\beta $$ .

Interfacing with PHCpack

This Macaulay2 package provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The package also provides a fast mixed volume computation, the ability to filter solutions, homotopy path tracking, and a numerical irreducible decomposition method. As the size of many problems in applied algebraic geometry often surpasses the capabilities of symbolic software, this package will be of interest to those working on problems involving large polynomial systems. NUMERICAL HOMOTOPY CONTINUATION. Many problems in applied algebraic geometry require solving, or counting the solutions of, a large polynomial or rational system. PHCpack is an inter- face the program PHCpack to one of several efficient polynomial system solvers that use numerical homotopy continuation methods (Li). The basic idea behind homotopy continuation is simple: to solve a polynomial system f(x) = 0, one first constructs a system g(x) = 0 that is easy to solve and then constructs a homotopy, H(x(t)) = (1 t)g(x)+ t f(x) in order to numerically track paths from known solutions of g (with t = 0) to the solutions of the target system f (with t = 1). Available since release 1.4 of Macaulay2 (M2), this package is motivated by (Ley) and uses the data types defined by Leykin in NAGtypes.m2. The main function of the package allows a Macaulay2 user to solve a system numerically through a blackbox solver, where the creation of the start system and homotopy continuation is done behind the scenes. The package also provides a fast mixed volume computation and allows the user to filter solutions, to track solution paths explicitly, and to perform numerical irreducible decompositions. This interface to PHCpack offers access to most of the functionality of the software, which has been serving as a development platform for many of the algorithms in numerical algebraic geometry (SVW). Computations in this paper were done with phc version 2.3.61 (version 1.0 was archived in (Ve1)). Since version 2.3.13, PHCpack contains MixedVol (GLW), and more recently added features are described in (Ve2). PHCpack can solve Laurent systems, so the package includes a method to convert a rational system to a Laurent polynomial system. The underlying polyhedral methods perform well on benchmark problems; in many of those, the mixed volume is computed essentially instantaneously. To use the methods in this package, the user must have the executable program phc available, preferably in the execution path. For more information on this, see the documentation for the package. Although PHCpack is open source, we follow the idea of OpenXM (MN + ) and require only that the

Algebraic boundaries of Hilbert’s SOS cones

AbstractWe study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

Numerically intersecting algebraic varieties via witness sets

The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F, for which A is an irreducible component of V(F), a generic linear space L of complementary dimension to A, and a numerical approximation to the set of witness points, L∩A. Given F, methods exist for computing a numerical irreducible decomposition, which consists of a collection of witness sets, one for each irreducible component of V(F). This paper concerns the more refined question of finding a numerical irreducible decomposition of the intersection A∩B of two irreducible algebraic sets, A and B, given a witness set for each. An existing algorithm, the diagonal homotopy, computes witness point supersets for A∩B, but this does not complete the numerical irreducible decomposition. In this paper, we use the theory of isosingular sets to complete the process of computing the numerical irreducible decomposition of the intersection.

Numerical algebraic geometry: a new perspective on gauge and string theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

Certified Numerical Homotopy Tracking

Given a homotopy connecting two polynomial systems, we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with an implementation in the numerical algebraic geometry package Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair.

A class of bilinear models for the optimization of energy conversion networks

Using principles from finite-time thermodynamics, this paper introduces a class of bilinear models suitable for financial optimization and risk management of commodity energy conversion processes. This class of high-level models is intended to predict the efficiency of energy conversion in campus scale utilities with complex energy requirements, multiple fuel sources, and significant operational flexibility. The models incorporate first and second law thermodynamic considerations. The bilinear character of these models derives from the multiplicative coupling between entropy flux and temperature driving forces that are intrinsic to the second law. Economic optimization of these models yields non-convex bilinear optimization problems. Special cases for optimal operation reduce to generalized eigenvalue problems, or more generally complex rank constraints that can be solved using numerical algebraic geometry. For general use, we propose a computational strategy based on a linear outer approximation coupled with a branch and bound methodology to reduce the search region. We demonstrate modeling and optimization using several example problems drawn from the existing literature on finite-time thermodynamics.

Toward a Salmon Conjecture

Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3×3×4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9. Combined with results of Landsberg and Manivel, this furnishes a computational solution of an open problem in algebraic statistics, namely, the set-theoretic version of Allman’s salmon conjecture for 4×4×4 tensors of border rank 4. A proof without numerical computation was given recently by Friedland and Gross.

Numerical-symbolic exact irreducible decomposition of cyclic-12

AbstractIn 1992, Göran Björck and Ralf Fröberg completely characterized the solution set of cyclic-8. In 2001, Jean-Charles Faugère determined the solution set of cyclic-9, by computer algebra methods and Gröbner basis computation. In this paper, a new theory in matrix analysis of rank-deficient matrices together with algorithms in numerical algebraic geometry enables us to present a symbolic-numerical algorithm to derive exactly the defining polynomials of all prime ideals of positive dimension in primary decomposition of cyclic-12. Empirical evidence together with rigorous proof establishes the fact that the positive-dimensional solution variety of cyclic-12 just consists of 72 quadrics of dimension one.

Mechanism mobility and a local dimension test

Abstract The mobility of a mechanism is the number of degrees of freedom (DOF) with which it may move. This notion is mathematically equivalent to the dimension of the solution set of the kinematic loop equations for the mechanism. It is well known that the classical Grubler–Kutzbach formulas for mobility can be wrong for special classes of mechanisms, and even more refined treatments based on displacement groups fail to correctly predict the mobility of so-called “paradoxical” mechanisms. This article discusses how recent results from numerical algebraic geometry can be applied to the question of mechanism mobility. In particular, given an assembly configuration of a mechanism and its loop equations, a local dimension test places bounds on the mobility of the associated assembly mode. A publicly available software code makes the idea easy to apply in the kinematics domain.

21st Century Kinematics: Synthesis, Compliance, and Tensegrity

As we move into the second decade of the 21st century, we can identify three research trends that we can expect to persist into the future. They are the analysis and synthesis of (i) spatial mechanisms and robotic systems, (ii) compliant linkage systems, and (iii) tensegrity and cable-driven systems. In each case, we find that researchers are formulating and solving polynomial systems of total degrees that dwarf those associated with major kinematics problems of the previous century.

Numerical algebraic geometry and algebraic kinematics

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

Efficient path tracking methods

Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for using Runge-Kutta methods in conjunction with adaptive precision are provided. These methods have been implemented in the Bertini software package, and several examples are described.

Numerical Algebraic Geometry for Optimal Control Applications

A new technique for solving polynomial nonlinear constrained optimal control problems is presented. The problem is reformulated into a parametric optimization problem, which in turn is solved in a two-step procedure. First, in a precomputation step, the equation part of the corresponding first order optimality conditions is solved for a generic value of the parameter. Relying on the underlying algebraic geometry, this first solution makes it possible to solve efficiently and in real time the corresponding optimal control problem at the measured parameter value for each subsequent time step. This approach has a probability one guarantee of finding the global optimal solution at each step. Controller synthesis for two applications from the area of power electronics featuring a dc-ac converter and a dc-dc converter are discussed to motivate the proposed approach.