Charles Wampler General Motors R&D Center (Adjunct, Univ. Notre Dame) Including joint work with

1. From Isolated Points to Positive Dimensions & Back Again:A Brief History of Numerical Algebraic Geometry Charles Wampler General Motors R&D Center (Adjunct, Univ. Notre Dame) Including joint work with Andrew Sommese, University of Notre Dame Jon Hauenstein, University of Notre Dame

2. 42 distinct co-authors

3. Outline • Introduction to continuation • Historical timeline • Algorithms for zero-dimensional solution sets • Algorithms for positive-dimensional solution sets • Creation of numerical algebraic geometry • Built on methods for zero-dimensional solving • Numerical irreducible decomposition • Intersecting algebraic sets • Diagonal homotopy • New: Regeneration method • Extensions of numerical algebraic geometry • Extracting real points • Computing the genus of a curve • Exceptional sets via fiber products • Using positive-dimensional methods to find isolated solution sets • Equation-by-equation solving using regeneration

4. S0 S1 t=0 t t=1 Introduction to Continuation • Basic idea: to solve F(x)=0 (N equations, N unknowns) • Define a homotopy H(x,t)=0 such that • H(x,1) = 0 has a known isolated solutions, S1 • H(x,0) = F(x) • Track solution paths as t goes from 1 to 0 • Paths satisfy the Davidenko o.d.e. • (dH/dx)(dx/dt) + dH/dt = 0 • Endpoints of the paths are solutions of F(x)=0 • Let S0 be the set of path endpoints • Concerns • Points where dH/dx is singular • Turning points • Path crossings • Multiple roots • Path divergence • Will S0 contain all isolated solutions of F(x)=0? • What happens if V(F) has positive dimensional components? • What if we have N equations, n unknowns, N≠n? • Numerical accuracy, Computational efficiency

5. Basic Total-degree Homotopy To find all isolated solutions to the polynomial system F: CN CN, i.e., form the linear homotopy H(x,t) = (1-t)F(x) + tG(x)=0, where

6. Brief Timeline: Part 1 (Before Sommese) • Pre-computer age • Homotopy & continuity for existence proofs • 1960s: early computational methods • Real number field, heuristic methods • “Bootstrap method” in kinematics – Roth 1963 • 1970s, early 80s: Algorithmic methods • Complex number field, Geometric arguments • Fixed-point continuation (T.Y.Li 1976) • Specialization to polynomial systems • All isolated solutions, 1977 (Garcia & Zangwill; Drexler) • Total degree homotopy, 1978 (Chow, Mallet-Paret & Yorke) • Numerical path tracking (Allgower & Georg, book ’80) • Use of projective space (Wright, 1985) • Application to 6R robot kinematics (Tsai & Morgan, 1985) • Morgan’s book on polynomial continuation 1987

7. S0 S1 t=0 t t=1 Addressing the concerns Concerns • Points where dH/dx is singular • Turning points • Path crossings • Multiple roots • Path divergence • Will S0 contain all isolated solutions of F(x)=0? • Computational efficiency • Numerical accuracy • What happens if V(F) has positive dimensional components? • What if we have N equations, n unknowns, N≠n? complex homotopy 1977 projective space 1985 1977 Total degree 1978

8. 6R Serial-Link Inverse Kinematics Formulation Pieper, 1968 & earlier 32nd degree polynomial Duffy & Crane, 1980 16 solutions shown by continuation Tsai & Morgan, 1985 Total degree homotopy with 256 paths 16th degree polynomial Li & Liang, 1988 Kinematic Milestone

9. Brief Timeline: Part 2 (After Sommese) • Late 80s, early 90s: Use of algebraic geometry • Still finding all isolated solutions • Focus on reducing the number of paths • Homotopies adapted to the structure of the target system • Multihomogeneity & the “γ trick” (Morgan & Sommese, 1987) • Parameterized systems: f(x;p)=0 • Cheater’s homotopy (Li, Sauer & Yorke, 1988) • Coefficient parameter homotopy (Morgan & Sommese, 1989) • Polytope homotopies (a.k.a., polyhedral, mixed volume, or BKK) • Verschelde, Verlinden & Cools, ’94; Huber & Sturmfels, ’95 • T.Y. Li with various co-authors, 1997-present • Computing singular endpoints • Morgan, Sommese, & Wampler (1991,1992) • Kinematics applications • Tutorial for mechanical engineers (MSW 1990) • Complete solution of the 9-point four-bar, (MSW 1992)

10. The “γ trick” • Simple but useful consequence of algebraicity • Instead of the homotopy H(x,t) = (1-t)f(x) + tg(x) use H(x,τ) = (1-τ)f(x) + gτg(x)

11. Parameter Continuation initial parameter space target parameter space Start system easy in initial parameter space Root count may be much lower in target parameter space Initial run is 1-time investment for cheaper target runs

12. 9-Point Path Generation for Four-bars Problem statement Alt, 1923 Bootstrap partial solution Roth, 1962 Complete solution Wampler, Morgan & Sommese, 1992 m-homogeneous continuation 1442 Robert cognate triples Kinematic Milestone

13. Nine-point Four-bar summary • Symbolic reduction • Initial total degree ≈1010 • Roth & Freudenstein, tot.deg.=5,764,801 • Our reformulation, tot.deg.=1,048,576 • Multihomogenization 286,720 • 2-way symmetry 143,360 • Numerical reduction (Parameter continuation) • Nondegenerate solutions 4326 • Roberts cognate symmetry 1442 • Synthesis program follows 1442 paths

14. Parameter Continuation: 9-point problem 2-homogeneous systems with symmetry: 143,360 solution pairs 9-point problems*: 1442 groups of 2x6 solutions *Parameter space of 9-point problems is 18 dimensional (complex)

15. S0 S1 t=0 t t=1 Addressing the concerns • Points where dH/dx is singular • Turning points • Path crossings • Multiple roots: Singular endgames 1991-92 • Path divergence • Will S0 contain all isolated solutions of F(x)=0? • Computational efficiency • Ab initio: • Multihomogeneous 1987 • Polytopes 1994-present • Parameter homotopy 1989 • Numerical accuracy • What happens if V(F) has positive dimensional components? • What if we have N equations, n unknowns, N≠n?

16. Sommese & Wampler (FoCM 1995 , publ. 1996) Brief Timeline: Part 3 (Numerical Alg.Geom.) • “Numerical Algebraic Geometry” founded • Treatment of positive dimensional sets by slicing • Treatment of N≠n by randomization • Cascade algorithm for slicing • Sommese & Verschelde, 2000 • Numerical irreducible decomposition • Factoring by sampling & fitting (Sommese, Verschelde, & Wampler, 2001) • Coinage of “witness set” • Use of monodromy (SVW 2001 ) • Use of symmetric functions (SVW 2002) • Intersection of components • Diagonal homotopy (SVW 2004) • Book by Sommese & Wampler, 2005 • Local methods • Deflation of μ>1 isolated points (Leykin, Verschelde, & Zhao, 2006) • Multiplicity structure (Dayton & Zeng 2005, Bates, Sommese, Peterson 2006) • Deflation of nonreduced positive dimensional components (S&W 2005) • Curves • Real curves (Lu, Bates, S&W, 2007), Curve genus (Bates, Peterson, S&W, preprint) • Equation-by-equation solving • Using diagonal homotopy (SVW 2008) • Using regeneration (Hauenstein, S, & W, in preparation) Up to positive dimensions And back again!

17. Slicing & Witness Sets • Slicing theorem • Andegree N reduced algebraic set of dimension m in n variables hits a general (n-m)-dimensional linear space in N isolated points • Witness generation • Slice at every dimension • Use continuation to get sets that contain all isolated solutions at each dimension • “Witness supersets” • Irreducible decomposition • Remove “junk” • Monodromy on slices finds irreducible components • Linear traces verify completeness

18. Membership Test

19. Linear Traces Track witness paths as slice translates parallel to itself. Centroid of witness points for an algebraic set must move on a line.

20. Further Reading World Scientific 2005

21. Example: Griffis-Duffy Platform Special Stewart-Gough platform Degree 28 motion curve (in Study coordinates) • if legs are equal & plates congruent: • factors as 6+(6+6+6)+4 Studied by: Husty & Karger, 2000

22. Real Points on a Complex Curve • Go to Griffis-Duffy movie…

23. S0 S1 t=0 t t=1 Addressing the concerns • Points where dH/dx is singular • Turning points • Path crossings • Multiple roots • Path divergence • Will S0 contain all isolated solutions of F(x)=0? • What happens if V(F) has positive dimensional components? • What if we have N equations, n unknowns, N≠n? • Computational efficiency • Isolated points • Positive dimensional components • Cascade goes like total degree • Working equation-by-equation helps! (esp., regeneration) • Numerical accuracy – a talk for another day Using traditional homotopies

24. Intersect Intersect Intersect • Special case: • N=n • nonsingular solutions only • results are very promising Equation-by-Equation Solving N equations, n variables f1(x)=0  Co-dim 1 f2(x)=0  Co-dim 1 Co-dim 1,2 f3(x)=0  Co-dim 1 Co-dim 1,2,3 Theory is in place for μ>1 isolated and for full witness set generation. Similar intersections Final Result Co-dim 1,2,...,N-1 Co-dim 1,2,...,min(n,N) fN(x)=0  Co-dim 1

25. Working Equation-by-Equation • Basic step

26. Regeneration: Step 1 Union of sets move slice dk times

27. Regeneration: Step 2 Linear homotopy

28. Software for polynomial continuation • PHC (first release 1997) • J. Verschelde • First publicly available implementation of polytope method • Used in SVW series of papers • Isolated points • Multihomogeneous & polytope method • Positive dimensional sets • Basics, diagonal homotopy • Hom4PS-2.0, Hom4PS-2.0para (recent release) • T.Y. Li • Isolated points: • Multihomogeneous & polytope method • Bertini (ver1.0 released Apr.20, 2008) • D. Bates, J. Hauenstein, A. Sommese, C. Wampler • Isolated points • Multihomogeneous • Positive dimensional sets • Basics, diagonal homotopy, & regeneration

29. Test Run 1: 6R Robot Inverse Kinematics *All runs in Bertini

30. Test Run 2: 9-point Four-bar Problem 1442 Roberts cognates

31. Test Run 3: Lotka-Volterra Systems • Discretized (finite differences) population model • Order n system has 8n sparse bilinear equations Work Summary + mixed volume Total degree = 28n Mixed volume = 24n is exact

32. Lotka-Volterra Systems (cont.) • Time Summary xx = did not finish All runs on a single processor

33. Algebraic vs. Geometric Approaches

34. Summary • Polynomials arise in applications • Especially kinematics • Continuation methods for isolated solutions • Highly developed in 1980’s, 1990’s • Numerical algebraic geometry • Builds on the methods for isolated roots • Treats positive-dimensional sets • Witness sets (slices) are the key construct • Regeneration: equation-by-equation approach • Uses slice moves to regenerate each new equation • Captures much of the same structure as polytope methods, without a mixed volume computation • Most efficient method yet for large, sparse systems • Thanks, Andrew, for 2 decades of fun collaboration!