REU 1992: Critical points of real polynomials (A. Durfee)

The REU project for 1992 directed by Alan Durfee continued the work of Durfee's 1989 REU group. The basic topic was the following: 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? A detailed description of this problem can be found at the 1989 site. The 1992 group, among other things, found a polynomial with an arbitrary number of local maxima and no other critical points. They also investigated and implemented computer algorithms for finding these critical points, and studied their graphical representation.

Student participants

Currently Thomas Feng is a graduate student in mathematics at Princeton University, email tfeng@math.princeton.edu, and Ian Robertson is a graduate student in mathematics at the University of Chicago, email ian@math.uchicago.edu. (The picture is of all three 1992 REU groups.)

Reports

The group produced the following reports:

Dana Fabbri and Silvia Rolloff, A graphical look at polynomials of two variables up to degree three at infinity

Abstract: A study of the behavior at infinity of polynomials with many pictures. The graphs are "scrunched" using the arctan function. (Hardcopy available from the Department of Mathematics, Mount Holyoke College, S. Hadley MA 01075)

Thomas Feng, Report

Abstract: This report discusses three topics: First, an efficient algorithm for finding simultaneous roots of two real polynomial equations in the plane; second, an improved upper bound on the number of roots in a special case; and third, a method for combining critical points. (postscript)

Ian Robertson, A polynomial with n maxima and no other critical points

Abstract: This short paper gives an explicit real polynomial of two variables with an arbitrary number of local maxima and no other critical points. Previously known (an easier to construct) were a polynomial with an arbitrary number of local extrema and no other critical points, a polynomial with two local maxima and no other critical points, and an analytic function with an arbitrary number of local maxima. (postscript)

Ian Robertson, Pedersen's algorithm for counting roots of polynomials in two variables

Abstract: An outline is given of an algorithm developed by Paul Pedersen for counting (without multiplicity) the number of real roots of a discrete algebraic set that lie within a region defined by the positivity of some polynomial. The algorithm can be applied to any dimension, though the description here will be confined to dimension two. (postscript)

Ian Robertson, Report

Abstract: An overview of Robertson's activities at the REU. (postscript)


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