A Numerical Approach to Real Algebraic Curves with the Wolfram Language

Bridging the gap between the sophisticated topic of real algebraic curve theory and on-the-spot computation and visualization of real algebraic curves, author Barry H. Dayton uses the Wolfram Language to explore and analyze real curves that often do not have rational points on them. In classical texts, analysis of these types of real curves was only really possible in the theoretical sense, but the Wolfram Language's ability to work with machine numbers, both in calculations and in detailed plots, enables accurate analysis of extremely complicated curves. This book is intended for those with some understanding of calculus and partial derivatives and with basic knowledge of the Wolfram Language.

Available for Kindle Fire, iPad or other tablets. Explore the author's code in his original Wolfram Computational Notebooks, and learn more about Wolfram technology at www.wolfram.com/language.

Contents

• Introduction to Numerical Plane Curves
• Classical vs. Numerical Perspective
• Special Points and Lines
• Gauss Decomposition
• A Classical Interlude
• Fractional Linear Transformations
• Asymptotes, Conics, and Cubics
• Global Plots
• Harnack's Theorem and Newton Hyperbolas
• Final Examples
• Appendix 1: Topics in Numerical Linear Algebra
• Appendix 2: Global Functions

About the Author

Barry H. Dayton received his PhD in mathematics with specialization in algebraic topology from the University of Southern California in 1970. From 1970 to 1990 his published research was in the abstract areas of K-theory, commutative algebra and algebraic geometry. Starting in the 1990s, he became interested in using numerical methods, culminating in his collaboration on multiplicity and deflation in numerical analysis with Zhonggang Zeng in 2005–2011. Since then, he has been working in the field of numerical algebraic geometry. He briefly taught at Harvey Mudd College but has mostly taught at Northeastern Illinois University where he is currently Professor Emeritus.