An intrinsic homotopy for intersecting algebraic varieties
Andrew J. Sommese, Jan Verschelde, Charles W. Wampler
TL;DR
The paper tackles computing the irreducible decomposition of $A\cap B$ for irreducible $A,B\subset\mathbb{C}^k$ by representing components with witness sets and using a diagonal projection via $X=A\times B$ and the diagonal $\Delta$. It introduces intrinsic coordinates to rewrite the diagonal homotopy, reducing the ambient variable count from $M=3k-a$ to $m=2k-\deg(A)-\deg(B)$ and enabling precomputation of linear parts, which yields substantial speedups in the path-tracking and linear solves. The authors develop intrinsic start and cascade homotopies, prove their equivalence with the extrinsic formulations for generic parameters, and present a three-stage algorithm with practical subroutines. Numerical experiments demonstrate roughly a factor-two overall speedup, with significant reductions in the cost of linear solving and substantial savings in CPU time for the intrinsic formulation, while function evaluation remains a practical bottleneck.
Abstract
Recently we developed a diagonal homotopy method to compute a numerical representation of all positive dimensional components in the intersection of two irreducible algebraic sets. In this paper, we rewrite this diagonal homotopy in intrinsic coordinates, which reduces the number of variables, typically in half. This has the potential to save a significant amount of computation, especially in the iterative solving portion of the homotopy path tracker. There numerical experiments all show a speedup of about a factor two.