MCS 595 Graduate Seminar, Fall 2003

This semester, we meet weekly in 712 SEO on Thursdays at 11:00. The theme of our seminar is "solving polynomial systems".
  1. Thu 28 Aug: "The Numerical Solution of Polynomial Systems arising in Engineering and Science". Explanation of table of contents in developing book project joint with Andrew Sommese and Charles Wampler shows default flow of seminar, eventually with some extra topics.
  2. Thu 4 Sep: "Introduction to Numerical Homotopy Continuation". To illustrate the regularity issues, download the Maple worksheet or look at the html version.
  3. Thu 11 Sep: "Path Tracking". We saw the role of the gamma constant to prevent paths running off to infinity and discussed predictor-corrector methods with adaptive step size control.
  4. Thu 18 Sep: "accessibility". Using resultants we showed that all isolated solutions can be reached by a total degree homotopy.
  5. Thu 25 Sep: "multi-homogeneous homotopies". We motivated the use of Bezout's theorem in multi-projective space via the eigenvalue problem and showed the combinatorial root count via linear-product start systems. This multi-homogeneous Bezout number is computed as a (generalized) permanent.
  6. Thu 2 Oct: "general linear-product start systems". We can generalize the multi-homogeneous homotopies by modeling the product structures by general arrays of sets, instead of using the same partition for every equation. While these homotopies predate (and are to a great extent surpassed by) the polyhedral methods, we introduced in this lecture the exploitation of symmetry and the Minkowski sum of Newton polytopes.
  7. Thu 9 Oct: "introduction to Newton polytopes". We looked at a very sparse class of systems: polynomials with in each equation exactly two monomials with nonzero coefficients. We call these systems "binomial systems". With unimodular transformations we showed that every binomial system has exactly as many regular roots as the determinant of the matrix of exponent vectors (taken in absolute value).
  8. Thu 16 Oct: "coefficient-parameter homotopy" by Anton Leykin. Systems arising in mechanical applications like the design of Stewart-Gough platforms have natural parameters, leading up to the so-called coefficient-parameter continuation.
  9. Thu 23 Oct: "introduction to polyhedral continuation". To avoid mixed volumes, we considered the gluing of positive real solutions of a polynomial in one variable. We can compute the solutions numerically with Newton's method or represent the solutions as fractional power series, which we also can compute with Newton's method. The generalization of the gluing leads to the patchworking method of Viro to construct algebraic curves with a prescribed topology.
  10. Thu 30 Oct: "the Cayley trick and Minkowski's theorem". The Cayley trick allows to write a resultant of a system as a discriminant of one polynomial using extra variables. With a geometric version of the Cayley trick we can visualize Minkowski's theorem which defines mixed volumes. Regular subdivisions lead to polyhedral homotopies.
  11. Thu 6 Nov: "the theorems of Bernshtein". Bernshtein proved in his landmark paper that the mixed volume is a root count for generic systems (Theorem A) and -- in the nongeneric case -- that solutions at infinity occur as roots of face systems (Theorem B). We stated these theorems precisely, sketched the main proof ideas and its practical implications.
  12. Thu 13 Nov: Olga Kashcheyeva on blowups I
  13. Thu 20 Nov: Olga Kashcheyeva on blowups II
  14. Thu 27 Nov: we do not meet because of Thanksgiving.
  15. Thu 4 Dec: Yusong Wang on Pieri homotopies
  16. Wed 10 Dec: Ailing Zhao on Newton with deflation for singularities