Solving Polynomials II

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" summary

Homotopy continuation constructs explicit start systems and traces solution paths so each isolated root receives exactly as many paths as its multiplicity, using a Bézout homotopy (6^5 = 7,776 start paths) and projective compactification to turn divergent paths finite. Singular endpoints use endgames (Puiseux series or contour integrals) and deflation by adding Jacobian minors to restore quadratic Newton convergence. Exploiting structure yields 3264 conics or the mixed volume (BKK) bound instead of crude overcounts.

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" chapters (20)

1. 0:00Introduction
2. 1:07Homotopy setup for solutions
3. 3:15Real coefficients and gamma
4. 7:11Solution paths in homotopy
5. 12:58Paths diverging to infinity
6. 17:14Compactification and projective space
7. 19:23Singular endpoints
8. 26:12End games and singular points
9. 30:41Newton's method convergence
10. 35:41Bertini's theorem for overdetermined
11. 40:31Multiplicity and F2 G example
12. 43:49Steiner's problem and conics
13. 47:25Exploiting system structure
14. 50:01Multihomogeneous and polytope homotopies
15. 52:08Determinantal systems
16. 56:13Newton and fixed-point homotopies
17. 59:19Coefficient parameter homotopy
18. 1:02:34Smale's 17th problem
19. 1:09:32Complexity and numerical methods
20. 1:15:22Q&A