Barry Dayton's Space

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

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For secure accesss to this site use https://barryhdayton.space/ In any case no senstive information is collected by or transmitted from this site.
Email: barry@barryhdayton.us Personal Note: I am now living in Ridgefield Connecticut.
Google Scholar Page Theory of Equations Book
More on Witt Vectors:
Papers with C.A. Weibel: SK_{1}Module Structures on the Hochschild and cyclic homology of graded rings
(with Z.Zeng and T.Y. Li) Multiple Zeros of Non-linear Systems Algebraic Foundation of Local Multiplicity
Configurations of Lines AG11 Numerical Algebraic Geometry via Numerical Polynomial Algebra AN12 Numerical Algebraic Geometry via Macaulay's Perspective
"Harnessing Natural Mosaics:
Antibody-Instructed, Multi-Envelope HIV-1 Vaccine Design"
Original material in this website is covered by |
My goal is an exposition of classical algebraic geometry (eg. 19th century) using Mathematica code. Abstract 20th century algebraic geometry, while interesting, is meaningless without a good classical background. Most current expositions of algorithms feature pseudo-code with an obsession in complexity, a totally useless concept. By using actual code the reader with access to Mathematica can directly verify my work and the speed of the algorithms.
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 The bulk of the book should be accessible to individuals who have had a standard pre-calculus course and have a working knowledge of the Wolfram Language. 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.
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.
I first discuss Then I 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 then be established. The last part covers, 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. NEW This story covers naive implicit surfaces and parametric surfaces in reall projective three space. The second edition is now available. It is divided into 3 parts, the first part gives basic definitions and a preview of the whole book. The second part gives details on quadric surfaces, cubic surfaces and some higher dimensional surfaces, especialy the torus. The third part concerns the topology of hyperboloids and tori with an application to the topology of complex projective plane conic and cubic curves. Here are several recent papers of mine. Note on Intersection of Rational Curves (PDF) and Interpolating Rational Normal Curves. See also my earlier paper Degree vs Dimension for rational parametric curves in the Mathematica Journal. NEW I have recently posted several self contained summaries of some of the material in my books in the Wolfram Community. These are available in web readable formats as well as Mathematica Notebooks. Barry H Dayton Wolfram Community Home |