Barry Dayton's Space

This web space is primarily for my ongoing research and writing, although some of my older material is also included.

Other Stuff
Email: barry@barryhdayton.us

Google Scholar Page
MathSciNet Author

My Northeastern homepage

Theory of Equations Book
2002 edition

Witt vectors and Necklaces
Short version
Long version

More on Witt Vectors:
The Picard Group of a reduced G-Algebra
See also B.Singh On Subintegrality and M.Vitulli Weak Normality and Seminormality

Papers with C.A. Weibel:
Naturality of Pic, SK0 and SK1
Module Structures on the Hochschild and cyclic homology of graded rings

Multiplicity

(with Z.Zeng and T.Y. Li) Multiple Zeros of Non-linear Systems

Algebraic Foundation of Local Multiplicity

Recent Talks and Papers

Approximate Local Rings 2006

Configurations of Lines
See Also Analysis of Bertini Experiments and Appendix

FCRC11-SNC11 Duality Talk

AG11 Numerical Algebraic Geometry via Numerical Polynomial Algebra

AN12 Numerical Algebraic Geometry via Macaulay's Perspective

Quadratic Surface Intersection Curves


Original material in this website is covered by

except for material in the sub-directory curvebook which is copyrighted by Wolfram Media and Barry H Dayton.

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.

Volume 2: A Numerical approach to Real Algebraic
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.

More...