Barry Dayton's Space
This web space is primarily for my ongoing research and writing, although some of my older material is also included.
Theory of Equations Book
Papers with C.A. Weibel:
Recent Talks and Papers
AG11 Numerical Algebraic Geometry via Numerical Polynomial Algebra
AN12 Numerical Algebraic Geometry via Macaulay's Perspective
Original material in this website is covered by
My current project is a 2 volume book on numerical algebraic curve theory based on Mathematica. The first volume of this book has now been published by Wolfram Media in several versions, Kindle and paperback editions available at Amazon.com and a Wolfram language notebook version for use with Mathematica or the Wolfram CDF reader. In addition an article length summary of this book has been published in The Mathematica Journal.
Volume 1: A Numerical approach to Real Algebraic
Plane Curves with the Wolfram Language
The main point of this book is to show that appropriate computer software can make this subject accessible to those who do not have the patience to master the big abstract theorems that usually define the field of abstract algebraic geometry. That doesn't mean we can ignore those theorems but does mean that we can concentrate on the application of these theorems rather than their development. To some extent this book was motivated by Shreeram Abhyankar's text Algebraic Geometry for Scientists and Engineers which, in my mind, missed the target audience by failing to illustrate the abstract ideas developed. I hope this book proves more interesting and readable to this audience.
The bulk of the book should be accessible to individuals who have enough familiarity with calculus to know what a partial derivative is. Numerical linear algebra is extensively used in the book but is hidden in the algorithms. An appendix is provided for those with a basic knowledge of linear algebra who want to learn more about this aspect.
Space Curves via Mathematica
Space curves in n-dimensional affine or projective space present a major challenge that we did not need to deal with for plane curves, instead of a single equation a system of n-1 or more equations are needed. This system is far from unique and, in many cases, may be over-determined. Since I allow numerical coefficients and, in general, numerical over-determined systems are to be avoided, specical methods must be used.
The first two Chapters will discuss naive space curves in 3-space, that is, curves defined by a system of two equations in 3 variables. Not only is this case familiar to what we learned in multivariable calculus but many methods in the plane curve case such as critical points and path tracing are still available.
The next few chapters develop the methods from numerical linear algebra that will be needed, Macaulay and Sylvester matrices and duality. The basic definitions and techniques for the general case will be established.
The last few chapters will cover, as examples, various situations I have recently presented in papers and talks. These are listed at the end of the left sidebar of this webpage.