Introduction to Computational Algebraic Geometry Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science http://www.math.uic.edu/˜jan [email protected]UIC ASCEND Workshops, 24 July 2008 Jan Verschelde (UIC) Computational Algebraic Geometry 24 July 2008 1 / 22
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Introduction to Computational Algebraic Geometry …homepages.math.uic.edu/~jan/Talks/introcage.pdfIntroduction to Computational Algebraic Geometry Jan Verschelde University of Illinois
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Introduction to Computational Algebraic Geometry
Jan Verschelde
University of Illinois at ChicagoDepartment of Mathematics, Statistics, and Computer Science
Jan Verschelde (UIC) Computational Algebraic Geometry 24 July 2008 1 / 22
Computational Algebraic Geometryan introduction to a modern mathematical discipline
The big picture:
What is algebraic geometry?
Algebraic geometry studies solutions of polynomial systems.
Polynomial systems occur in a wide variety of applications.
Using computers to discover theorems.
Computer algebra software offers implementations of algorithmsto solve polynomial systems.
We will use SAGE, an open source software system.
Problem of today:
How do two circles intersect?
Jan Verschelde (UIC) Computational Algebraic Geometry 24 July 2008 2 / 22
Outline
1 SAGE: Software for Algebra and Geometry ExperimentationTry it online!
2 An Intersection Problemplotting and solving specific instanceslooking at the general problem formulation
3 Determinants, Resultants and DiscriminantsJacobian matrices and singular solutionseliminating variables with resultantscomputing discriminants using resultants
Jan Verschelde (UIC) Computational Algebraic Geometry 24 July 2008 3 / 22
Using SAGESoftware for Algebra and Geometry Experimentation
SAGE is open source mathematical software1 compilation both of original Python, C, C++, and SageX code2 interfaces to computational algebraic geometry software: Singular3 the GUI is your web browser, try it before installation
Three steps to getting started:1 Go to http://www.sagemath.org
2 click on Try it online!3 Sign up for a new SAGE notebook account.
Jan Verschelde (UIC) Computational Algebraic Geometry 24 July 2008 4 / 22
Plotting Two CirclesConsider two circles, how do they intersect?
Geometric interpretation:� the discriminant gives the relation between center � c � 0 � andradius r of the second circle for which the solutions are singular, i.e.:
1 double solutions: circles touch each other2 a solution set: overlapping circles
Jan Verschelde (UIC) Computational Algebraic Geometry 24 July 2008 18 / 22
Factoring the Discriminantto simplify the condition on the parameters
To factor the discriminant, we must convert to an element of the ring R.