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MY RESEARCH RELATED LINKS	Barry H. Dayton Professor Emeritus	MY OTHER WEB PAGES
Theory of Equations Book Permutations and Necklaces Approximate local Rings (Paper presented at Notre Dame AMS meeting, April 9, 2006) Numerical Local Rings of Analytic Systems (January 2008) Talk at ACA 2007 Local Solution of Analytic Systems Talk at SNC 2007 Numerical Local Rings Talk at Depaul AMS Meeting, October 2007 Multiple Zeros of Analytic Systems Preprint August 2008 Numerically generic Unions of Lines Analysis of Bertini Experiments Data for Bertini Experiments Preprint January 2011 Algebraic Foundation of Local Multiplicity Preprint April 2011 Numerical Calculation of H-bases for Positive Dimensional Varieties SNC 2011 Talk on Numerical Calculation of H-bases... at ACM-FCRC conference, San Jose, CA, June 2011 (PDF slides) AG11 Talk on Numerical Algebraic Geometry at SIAM AG11 conference, Raleigh NC, October 2011 (PDF slides) Michigan Computational Algebraic Geometry talk on global duals at MCAG12, Oakland University, May 2012 (PDF slides). SIAM talk on Numerical Algebraic Geometry via Macaulay at Numerical Methods for Polynomial Systems, AN12, Minneapolis, July 2012 (PDF slides). Zip file for referees of QSIC paper	Lines in a cubic Surface In 1849 Salmon and Caley discovered that there are exactly 27 straight lines contained in a non-singular complex projective cubic in 3-space. Unfortunately it is not easy to show these in Euclidean 3-space, some of these lines may lie in the plane at infinity and most of these lines generally are complex. For example, the Fermat cubic specializes to the cubic x3+y3+z3=1 and while none of the lines are in the plane at infinity only 3 are real. In 1856 in an attempt to clarify the Salmon-Caley result Ludwig Schlaefli gave a construction of 12 lines that would lie in a cubic. The resulting set of lines is now known as "Schlaefli's double 6." It is easy from these to find the remaining 15 lines. These lines can be chosen, with care, to be real. Hilbert included this construction in his popular book on geometry "Geometry and the Imagination" written with S. Cohn-Vossen in 1932. This book is presently available from the American Mathematical Society www.ams.org/bookstore in English. While the method is nice in theory it may not be possible to actually find integer polynomial equations for the lines and the containing cubic surface in order to graph. If you are willing to allow decimal coefficients I explain how to find the actual equations in the projective case in my paper Numerically generic Unions of Lines and one can easily specialize to real 3-space. The picture above is a cubic surface containing a double 6, note 6 of the lines are red and 6 are black, none of the red (or black) lines intersect other red (resp. black) lines but each red line intersects 5 of the black lines and conversely. These calculations and the picture where made using Mathematica. The paper Numerical Calculation of H-bases for Positive Dimensional Varieties discusses lines in the cubic from an affine (non-projective) point of view.	Page for my Students Oakton Adjunct Faculty Assoc. Park Ridge Penguins Swim Team Lake Michigan Diary
	Home Page of Barry H Dayton Professor Emeritus Northeastern Illinois University Chicago, IL 60625-4699, USA B-Dayton@neiu.edu