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 circle. There are two elementary results: First, Bezout's theorem gives that m + n + s is less than or equal to d. 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.

picture of students The student participants were:

This research project was continued by the 1992 REU group. It also led to two papers produced by Durfee. The 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 Topology. 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 birthday.

[graph of a polynomial whose only critical points are a maximum and a minimum] 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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