Solving Polynomials I

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Compares symbolic elimination (Gröbner/resultant reductions to easier systems) with numeric homotopy continuation and path tracking, using homotopy continuation from an easy g(x)=0 to f(x)=0 and adaptive precision to avoid singular t-values; certifies convergence locally with alpha theory (Newton quadratic convergence and alphaCertified). Gives concrete bounds: Bézout and multi‑homogeneous counts, explains a 3×3 matrix‑multiplication setup with 729 polynomials (no solutions for r≤18, solutions for r≥23, unknown 19–22), and shows resultant reductions that drop a 4→3 root count.

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" chapters (13)

1. 0:00Start
2. 0:59Symbolic Methods
3. 2:55Linear System of Equations
4. 4:44Numerical Algebraic Geometry
5. 8:08Non Singular Solutions
6. 9:35Implicit Function Theorem
7. 22:37The Cubic Formula
8. 26:203 by 3 Matrix Model
9. 31:48Numerical Methods
10. 36:40Prove Quadratic Convergence
11. 44:08Fundamental Theorem of Algebra
12. 47:42Example of Using Basis Theorem
13. 59:15Degree of X