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.

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. The first, The index of

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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