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
 Dana Fabbri, University of Massachusetts '93
 Thomas Feng, Yale University '93
 Ian Robertson, Oberlin College '93
 Sylvia Rolloff, Mount Holyoke College '93
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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