1. Komplexe Analysis, Oberwolfach, August 31, 2006 Exceptional Sets and Fiber Products Andrew Sommese University of Notre Dame Charles Wampler General Motors R&D Center

2. Komplexe Analysis, Oberwolfach, August 31, 2006 2 Reference on Numerical Algebraic Geometry up to 2005: A.J. Sommese and C.W. Wampler, Numerical solution of systems of polynomials arising in engineering and science, (2005), World Scientific Press. Recent articles are available at www.nd.edu/~sommese

3. Komplexe Analysis, Oberwolfach, August 31, 2006 3 Overview A Motivating Problem Exceptional Sets and Fiber Products Numerical Algebraic Geometry Isolated Solutions Positive Dimensional A Case Study

4. Komplexe Analysis, Oberwolfach, August 31, 2006 4 A Motivating Problem and an Approach to It This is joint work with Charles Wampler. The problem is to find the families of overconstrained mechanisms of specified types.

5. Komplexe Analysis, Oberwolfach, August 31, 2006 5

6. 6 for i from 0 to 5 Inverse Displacement Problem: determine the L i from (p,R) Forward Displacement Problem: determine (p,R) from the L i

7. Komplexe Analysis, Oberwolfach, August 31, 2006 7 If the lengths of the six legs are fixed the platform robot is usually rigid. Husty and Karger made a study of exceptional lengths when the robot will move: one interesting case is when the top joints and the bottom joints are in a configuration of equilateral triangles.

8. Komplexe Analysis, Oberwolfach, August 31, 2006 8

9. 9 Another Example

10. Komplexe Analysis, Oberwolfach, August 31, 2006 10 Overconstrained Mechanisms

11. Komplexe Analysis, Oberwolfach, August 31, 2006 11 To automate the finding of such mechanisms, we need to solve the following problem: Given an algebraic map p between irreducible algebraic affine varieties X and Y, find the irreducible components of the algebraic subset of X consisting of points x with the dimension of the fiber of p at x greater than the generic fiber dimension of the map p.

12. Komplexe Analysis, Oberwolfach, August 31, 2006 12 One Approach A method to find the exceptional sets A.J. Sommese and C.W. Wampler, Exceptional sets and fiber products, preprint. A method to solve large systems with few solutions A.J. Sommese, J. Verschelde and C.W. Wampler, Solving polynomial systems equation by equation, preprint.

13. Komplexe Analysis, Oberwolfach, August 31, 2006 13 Exceptional Sets

14. Komplexe Analysis, Oberwolfach, August 31, 2006 14 A Simple Example

15. Komplexe Analysis, Oberwolfach, August 31, 2006 15 Setup

16. Komplexe Analysis, Oberwolfach, August 31, 2006 16 Problem Statement

17. Komplexe Analysis, Oberwolfach, August 31, 2006 17 For

18. Komplexe Analysis, Oberwolfach, August 31, 2006 18 Fiber Products

19. Komplexe Analysis, Oberwolfach, August 31, 2006 19

20. Komplexe Analysis, Oberwolfach, August 31, 2006 20 Main Components

21. Komplexe Analysis, Oberwolfach, August 31, 2006 21 The Notion of a Main Component

22. Komplexe Analysis, Oberwolfach, August 31, 2006 22

23. Komplexe Analysis, Oberwolfach, August 31, 2006 23

24. Komplexe Analysis, Oberwolfach, August 31, 2006 24 It is easy to recognize main components. An irreducible subset W of is a main component if

25. Komplexe Analysis, Oberwolfach, August 31, 2006 25

26. Komplexe Analysis, Oberwolfach, August 31, 2006 26 A Bound

27. Komplexe Analysis, Oberwolfach, August 31, 2006 27 An Alternate Bound

28. Komplexe Analysis, Oberwolfach, August 31, 2006 28 Numerical Algebraic Geometry Isolated Solutions Positive Dimensional Solutions Bertini A Case Study

29. Komplexe Analysis, Oberwolfach, August 31, 2006 29 Computing Isolated Solutions Find all isolated solutions in of a system on n polynomials:

30. Komplexe Analysis, Oberwolfach, August 31, 2006 30 Solving a system Homotopy continuation is our main tool: Start with known solutions of a known start system and then track those solutions as we deform the start system into the system that we wish to solve.

31. Komplexe Analysis, Oberwolfach, August 31, 2006 31 Path Tracking This method takes a system g(x) = 0, whose solutions we know, and makes use of a homotopy, e.g., Hopefully, H(x,t) defines “paths” x(t) as t runs from 1 to 0. They start at known solutions of g(x) = 0 and end at the solutions of f(x) at t = 0.

32. Komplexe Analysis, Oberwolfach, August 31, 2006 32 The paths satisfy the Davidenko equation To compute the paths: use ODE methods to predict and Newton’s method to correct.

33. Komplexe Analysis, Oberwolfach, August 31, 2006 33 Solutions of f(x)=0 Known solutions of g(x)=0 t=0 t=1 H(x,t) = (1-t) f(x) + t g(x) x 3 (t) x 1 (t) x 2 (t) x 4 (t)

34. Komplexe Analysis, Oberwolfach, August 31, 2006 34 Newton correction prediction {  t x j (t) x* 0 1

35. Komplexe Analysis, Oberwolfach, August 31, 2006 35 Algorithms middle 80’s: Projective space was beginning to be used, but the methods were a combination of differential topology and numerical analysis with homotopies tracked exclusively through real parameters. early 90’s: algebraic geometric methods worked into the theory: great increase in security, efficiency, and speed.

36. Komplexe Analysis, Oberwolfach, August 31, 2006 36 Uses of algebraic geometry Simple but extremely useful consequence of algebraicity [A. Morgan (GM R. & D.) and S.] Instead of the homotopy H(x,t) = (1-t)f(x) + tg(x) use H(x,t) = (1-t)f(x) +  tg(x)

37. Komplexe Analysis, Oberwolfach, August 31, 2006 37 Genericity Morgan + S. : if the parameter space is irreducible, solving the system at a random points simplifies subsequent solves: in practice speedups by factors of 100.

38. Komplexe Analysis, Oberwolfach, August 31, 2006 38 Endgames (Morgan, Wampler, and S.) Example: (x – 1) 2 - t = 0 We can uniformize around a solution at t = 0. Letting t = s 2, knowing the solution at t = 0.01, we can track around |s| = 0.1 and use Cauchy’s Integral Theorem to compute x at s = 0.

39. Komplexe Analysis, Oberwolfach, August 31, 2006 39 Special Homotopies to take advantage of sparseness

40. Komplexe Analysis, Oberwolfach, August 31, 2006 40 Multiprecision Not practical in the early 90’s! Highly nontrivial to design and dependent on hardware Hardware too slow

41. Komplexe Analysis, Oberwolfach, August 31, 2006 41 Hardware Continuation is computationally intensive. On average: in 1985: 3 minutes/path on largest mainframes.

42. Komplexe Analysis, Oberwolfach, August 31, 2006 42 Hardware Continuation is computationally intensive. On average: in 1985: 3 minutes/path on largest mainframes. in 1991: over 8 seconds/path, on an IBM 3081; 2.5 seconds/path on a top-of-the-line IBM 3090.

43. Komplexe Analysis, Oberwolfach, August 31, 2006 43 Hardware Continuation is computationally intensive. On average: in 1985: 3 minutes/path on largest mainframes. in 1991: over 8 seconds/path, on an IBM 3081; 2.5 seconds/path on a top-of-the-line IBM 3090. 2006: about 10 paths a second on an single processor desktop CPU; 1000’s of paths/second on moderately sized clusters.

44. Komplexe Analysis, Oberwolfach, August 31, 2006 44 A Guiding Principle then and now Algorithms must be structured – when possible – to avoid paths leading to singular solutions: find a way to never follow the paths in the first place.

45. Komplexe Analysis, Oberwolfach, August 31, 2006 45 Continuation’s Core Computation Given a system f(x) = 0 of n polynomials in n unknowns, continuation computes a finite set S of solutions such that: any isolated root of f(x) = 0 is contained in S; any isolated root “occurs” a number of times equal to its multiplicity as a solution of f(x) = 0; S is often larger than the set of isolated solutions.

46. Komplexe Analysis, Oberwolfach, August 31, 2006 46 References A.J. Sommese and C.W. Wampler, Numerical solution of systems of polynomials arising in engineering and science, (2005), World Scientific Press. T.Y. Li, Numerical solution of polynomial systems by homotopy continuation methods, in Handbook of Numerical Analysis, Volume XI, 209-304, North-Holland, 2003.

47. Komplexe Analysis, Oberwolfach, August 31, 2006 47 Positive Dimensional Solution Sets We now turn to finding the positive dimensional solution sets of a system

48. Komplexe Analysis, Oberwolfach, August 31, 2006 48 How to represent positive dimensional components? S. + Wampler in ’95: Use the intersection of a component with generic linear space of complementary dimension. By using continuation and deforming the linear space, as many points as are desired can be chosen on a component.

49. Komplexe Analysis, Oberwolfach, August 31, 2006 49 Use a generic flag of affine linear spaces to get witness point supersets This approach has 19 th century roots in algebraic geometry

50. Komplexe Analysis, Oberwolfach, August 31, 2006 50 The Numerical Irreducible Decomposition Carried out in a sequence of articles with Jan Verschelde (Univiversity at Illinois at Chicago) and Charles Wampler (General Motors Research and Development) Efficient Computation of “Witness Supersets’’ S. and V., Journal of Complexity 16 (2000), 572-602. Numerical Irreducible Decomposition S., V., and W., SIAM Journal on Numerical Analysis, 38 (2001), 2022-2046.

51. Komplexe Analysis, Oberwolfach, August 31, 2006 51 An efficient algorithm using monodromy S., V., and W., SIAM Journal on Numerical Analysis 40 (2002), 2026-2046. Intersections of algebraic sets S., V., and W., SIAM Journal on Numerical Analysis 42 (2004), 1552-1571.

52. Komplexe Analysis, Oberwolfach, August 31, 2006 52 Symbolic Approach with same classical roots Two articles in this direction: M. Giusti and J. Heintz, Symposia Mathematica XXXIV, pages 216-256. Cambridge UP, 1993. G. Lecerf, Journal of Complexity 19 (2003), 564-596.

53. Komplexe Analysis, Oberwolfach, August 31, 2006 53 The Irreducible Decomposition

54. Komplexe Analysis, Oberwolfach, August 31, 2006 54 Witness Point Sets

55. Komplexe Analysis, Oberwolfach, August 31, 2006 55

56. Komplexe Analysis, Oberwolfach, August 31, 2006 56 Basic Steps in the Algorithm

57. Komplexe Analysis, Oberwolfach, August 31, 2006 57 Example

58. Komplexe Analysis, Oberwolfach, August 31, 2006 58

59. Komplexe Analysis, Oberwolfach, August 31, 2006 59 From Sommese, Verschelde, and Wampler, SIAM J. Num. Analysis, 38 (2001), 2022-2046.

60. Komplexe Analysis, Oberwolfach, August 31, 2006 60 Numerical issues posed by multiple components Consider a toy homotopy Continuation is a problem because the Jacobian with respect to the x variables is singular. How do we deal with this?

61. Komplexe Analysis, Oberwolfach, August 31, 2006 61 Deflation The basic idea introduced by Ojika in 1983 is to differentiate the multiplicity away. Leykin, Verschelde, and Zhao gave an algorithm for an isolated point that they showed terminated. Given a system f, replace it with

62. Komplexe Analysis, Oberwolfach, August 31, 2006 62 Bates, Hauenstein, Sommese, and Wampler: To make a viable algorithm for multiple components, it is necessary to make decisions on ranks of singular matrices. To do this reliably, endgames are needed.

63. Komplexe Analysis, Oberwolfach, August 31, 2006 63 Bertini and the need for adaptive precision Why use Multiprecision? to ensure that the region where an endgame works is not contained the region where the numerics break down;

64. Komplexe Analysis, Oberwolfach, August 31, 2006 64 Bertini and the need for adaptive precision Why use Multiprecision? to ensure that the region where an endgame works is not contained the region where the numerics break down; to ensure that a polynomial is zero at a point is the same as the polynomial numerically being approximately zero at the point;

65. Komplexe Analysis, Oberwolfach, August 31, 2006 65 Bertini and the need for adaptive precision Why use Multiprecision? to ensure that the region where an endgame works is not contained the region where the numerics break down; to ensure that a polynomial is zero at a point is the same as the polynomial numerically being approximately zero at the point; to prevent the linear algebra in continuation from falling apart.

66. Komplexe Analysis, Oberwolfach, August 31, 2006 66 Evaluation To 15 digits of accuracy one of the roots of this polynomial is a = 27.9999999999999. Evaluating p(a) to 15 digits, we find that p(a) = -2. Even with 17 digit accuracy, the approximate root a is a = 27.999999999999905 and we still only have p(a) = -0.01.

67. Komplexe Analysis, Oberwolfach, August 31, 2006 67 Wilkinson’s Theorem Numerical Linear Algebra Solving Ax = f, with A an N by N matrix, we must expect to lose digits of accuracy. Geometrically, is on the order of the inverse of the distance in from A to to the set defined by det(A) = 0.

68. Komplexe Analysis, Oberwolfach, August 31, 2006 68 One approach is to simply run paths that fail over at a higher precision, e.g., this is an option in Jan Verschelde’s code, PHC.

69. Komplexe Analysis, Oberwolfach, August 31, 2006 69 One approach is to simply run paths that fail over at a higher precision, e.g., this is an option in Jan Verschelde’s code, PHC. Bertini is designed to dynamically adjust the precision to achieve a solution with a prespecified error. Bertini is being developed by Daniel Bates, Jon Hauenstein, Charles Wampler, and myself (with some early work by Chris Monico).

70. Komplexe Analysis, Oberwolfach, August 31, 2006 70 Issues You need to stay on the parameter space where your problem is: this means you must adjust the coefficients of your equations dynamically.

71. Komplexe Analysis, Oberwolfach, August 31, 2006 71 Issues You need to stay on the parameter space where your problem is: this means you must adjust the coefficients of your equations dynamically. You need rules to decide when to change precision and by how much to change it.

72. Komplexe Analysis, Oberwolfach, August 31, 2006 72 The theory we use is presented in the article D. Bates, A.J. Sommese, and C.W. Wampler, Multiprecision path tracking, preprint. available at www.nd.edu/~sommese

73. Komplexe Analysis, Oberwolfach, August 31, 2006 73 Case Study: Alt’s Problem We follow

74. Komplexe Analysis, Oberwolfach, August 31, 2006 74 A four-bar planar linkage is a planar quadrilateral with a rotational joint at each vertex. They are useful for converting one type of motion to another. They occur everywhere.

75. Komplexe Analysis, Oberwolfach, August 31, 2006 75 How Do Mechanical Engineers Find Mechanisms? Pick a few points in the plane (called precision points) Find a coupler curve going through those points If unsuitable, start over.

76. Komplexe Analysis, Oberwolfach, August 31, 2006 76 Having more choices makes the process faster. By counting constants, there will be no coupler curves going through more than nine points.

77. Komplexe Analysis, Oberwolfach, August 31, 2006 77 Nine Point Path-Synthesis Problem H. Alt, Zeitschrift für angewandte Mathematik und Mechanik, 1923: Given nine points in the plane, find the set of all four-bar linkages, whose coupler curves pass through all these points.

78. Komplexe Analysis, Oberwolfach, August 31, 2006 78 First major attack in 1963 by Freudenstein and Roth.

79. Komplexe Analysis, Oberwolfach, August 31, 2006 79 D′D′ PjPj δjδj λjλj µjµj u b v C D x y P0P0 C′C′

80. Komplexe Analysis, Oberwolfach, August 31, 2006 80 PjPj δjδj µjµj b-δ j v C y P0P0 θjθj ye iθ j b v = y – b ve iμ j = ye iθ j - (b - δ j ) = ye iθ j + δ j - b C′C′

81. Komplexe Analysis, Oberwolfach, August 31, 2006 81 We use complex numbers (as is standard in this area) Summing over vectors we have 16 equations plus their 16 conjugates

82. Komplexe Analysis, Oberwolfach, August 31, 2006 82 This gives 8 sets of 4 equations: in the variables a, b, x, y, and for j from 1 to 8.

83. Komplexe Analysis, Oberwolfach, August 31, 2006 83 Multiplying each side by its complex conjugate and letting we get 8 sets of 3 equations in the 24 variables with j from 1 to 8.

84. Komplexe Analysis, Oberwolfach, August 31, 2006 84

85. Komplexe Analysis, Oberwolfach, August 31, 2006 85 in the 24 variables with j from 1 to 8.

86. Komplexe Analysis, Oberwolfach, August 31, 2006 86

87. Komplexe Analysis, Oberwolfach, August 31, 2006 87 Using Cramer’s rule and substitution we have what is essentially the Freudenstein-Roth system consisting of 8 equations of degree 7. Impractical to solve: 7 8 = 5,764,801solutions.

88. Komplexe Analysis, Oberwolfach, August 31, 2006 88 Newton’s method doesn’t find many solutions: Freudenstein and Roth used a simple form of continuation combined with heuristics. Tsai and Lu using methods introduced by Li, Sauer, and Yorke found only a small fraction of the solutions. That method requires starting from scratch each time the problem is solved for different parameter values

89. Komplexe Analysis, Oberwolfach, August 31, 2006 89

90. Komplexe Analysis, Oberwolfach, August 31, 2006 90 Solve by Continuation All 2-homog. systems All 9-point systems “numerical reduction” to test case (done 1 time) synthesis program (many times)

91. Komplexe Analysis, Oberwolfach, August 31, 2006 91