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".
- 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.
- Thu 4 Sep: "Introduction to Numerical Homotopy Continuation".
To illustrate the regularity issues, download
the Maple worksheet or look at
the html version.
- 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.
- Thu 18 Sep: "accessibility". Using resultants we showed that
all isolated solutions can be reached by a total degree homotopy.
- 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.
- 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.
- 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).
- 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.
- 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.
- 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.
- 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.
- Thu 13 Nov: Olga Kashcheyeva on blowups I
- Thu 20 Nov: Olga Kashcheyeva on blowups II
- Thu 27 Nov: we do not meet because of Thanksgiving.
- Thu 4 Dec: Yusong Wang on Pieri homotopies
- Wed 10 Dec: Ailing Zhao on Newton with deflation for singularities