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American Mathematical Monthly -November 2006

November 2006

Curves in Cages: An Algebro-Geometric Zoo
Gabriel Katz
The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families of cardinality d. The union of all lines from both families is called a cage, and the intersection X consists of d2 points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. We study the subsets A of the nodal set X such that any plane algebraic curve C of degree d that contains A must contain X as well. As a corollary, we obtain several generalizations of the famous Pascal theorem, generalizations that employ polygons (instead of hexagons as in Pascal's theorem) inscribed in quadratic curves. Our results are closely related to the classical theorems of Chasles and Bacharach. Recall that these theorems deal with families FA and FB of algebraic curves of subtly bounded degrees that contain complementary subsets A and B of X. Here the set X is a given complete intersection. Although the nodal sets X produced by our cages are quite special in comparison with more general complete intersections studied by Bacharach, the cages provide us with a much better grip on the combinatorics of subsets A of X with the property just described.


An Optimization Framework for Polynomial Zerofinders
Aaron Melman and Bill Gragg,
This article exhibits a correspondence between methods for the computation of the zeros of a polynomial and a constrained optimization problem. This sheds new light on several classical methods like Newton's and Laguerre's, and it provides a framework for the construction of new methods. It also allows one to obtain overshooting properties for these methods.


Monotonicity Rules in Calculus
Glen Anderson, Mavina Vamanamurthy, and Matti Vuorinen,,
A basic result of calculus states that if a function f is continuous on an interval [a,b] and has a positive derivative on (a,b), then f is increasing on [a,b]. This result is obtained easily by means of the Lagrange mean value theorem. If one is attempting to prove monotonicity of a quotient of two functions, the derivative of the quotient may often be quite messy and the process tedious. The authors list several sufficient conditions for the monotonicity of such quotients. In particular, they provide many examples from calculus of the use of the so-called L'Hospital Monotone Rule for monotonicity, which relies on the quotient of the derivatives instead of the (more complicated) derivative of the quotient.


As Algebra, So Poetry
Sarah Glaz and JoAnne Growney,
A translation of Ut Algebra Poesis by the Romanian poet Ion Barbu



Several Colorful Inequalities
Ingram Olkin and Larry Shepp,

Combinatorics of Barycentric Subdivision and Characters of Simplicial Two-Complexes
David F. Snyder

On Products of Euclidean Reflections
Thomas Brady and Colum Watt,

Another Short Proof of Descartes's Rule of Signs
Vilmos Komornik

The Evolution ofÂ… The Life and Work of Alexander Grothendieck
by Piotr Pragacz

Problems and Solutions


Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures.
By James Robert Brown
Reviewed by Charles R. Hampton

Philosophies of Mathematics.
By Alexander George and Daniel Velleman
Reviewed by Charles R. Hampton

Thinking about Mathematics: The Philosophy of Mathematics.
By Stewart Shapiro
Reviewed by Charles R. Hampton