REU 1989: Counting critical points of real polynomials in two variables (A. Durfee)
The REU project for 1989 directed by Alan Durfee investigated the
following topic: Given a real polynomial f(x,y) of two variables of
degree d , with only nondegenerate critical points, what possible
combinations of saddles, local maxima and local minima can occur?
This problem was suggested by V. I. Arnold especially for our group.
Let m be the number of maxima, n the number of
minima and s the
number of saddles, and let i be the index of grad f(x,y) about a large
There are two elementary results:
First, Bezout's theorem gives that m + n + s is less than or equal
Secondly, the index
theorem gives that |i| is at most d-1.
The group produced the paper Counting Critical Points of Real
Polynomials in Two Variables which was published in the American
Mathematical Monthly 100 (1993) 255-271.
The paper provides a detailed introduction to the problem. It also discusses
polynomials that factor completely into real linear factors and the
relation to Hilbert's Sixteenth Problem.
Some examples are given of polynomials with i > 1; these are
rather hard to find.
The main result of the paper is that if i>1, then the polynomial is
not generic in the sense that its partial derivatives have common
zeros on the line at infinity.
The student participants were:
- Nathan Kronenfeld, Harvard College '90
- Heidi Munson, St. Olaf College '90
- Jeff Roy, Harvard College '90
- Ina Westby, St. Olaf '90
This research project was continued by the 1992
REU group. It also led to two papers produced by Durfee.
first, The index of grad
f(x,y) uses techniques from algebraic geometry and topology to
produce a lower bound for the index i . This will appear in
The second, Five definitions of critical point
at infinity, is a survey of critical points at infinity for
complex polynomials of two variables. This will appear in the
proceedings of the Oberwolfach conference in honor of Brieskorn's 60th
Graph of the polynomial y^5 + x^2y^3 - y, which has a local maximum, a local minimum and no other critical points.
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