Math 480, Computational Algebraic Geometry - Revision history

Steven.Jackson at 17:23, 10 June 2012

2012-06-10T17:23:27Z

Steven.Jackson: Added link to git repository.

2012-06-10T17:21:20Z

Added link to git repository.

Patrick.Mclaren: Fixed google code typo

2012-06-04T17:22:13Z

Fixed google code typo

Steven.Jackson at 11:04, 3 May 2012

2012-05-03T11:04:47Z

Patrick.Mclaren: Question regarding polynomial implementation

2012-05-03T01:49:00Z

Question regarding polynomial implementation

Patrick.Mclaren: Added REU

2012-05-01T22:08:49Z

Added REU

Steven.Jackson at 20:44, 1 May 2012

2012-05-01T20:44:49Z

Patrick.Mclaren: Added source code repository.

2012-05-01T14:35:34Z

Added source code repository.

Patrick.Mclaren: Added history, course structure

2012-05-01T14:28:29Z

Added history, course structure

Patrick.Mclaren: Created page

2012-05-01T13:10:52Z

Created page

New page

Math 480 -- Computational Algebraic Geometry

Welcome to the wiki! Editing this page is exactly like editing [http://en.wikipedia.org/wiki/Main_page Wikipedia]. You may wish to see their help pages on [http://en.wikipedia.org/wiki/Help:Contents/Editing_Wikipedia editing] and on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].

;Course Textbook
The course mainly follows [http://www.cs.amherst.edu/~dac/iva.html Ideals, Varieties, and Algorithms] by Cox, Little, and O'Shea, however other textbooks can also provide supplementary material.

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 ; Code repository
-Please clone the course's [http://cartan.math.umb.edu/gitweb git repository] to your local machine and submit code assignments with <code>git push</code>.  (See [http://git-scm.com/book/en/Getting-Started-Installing-Git#Installing-on-Mac this link] for instructions on installing git, after which you might want to type <code>man gittutorial</code> from your machine's command line.)
+Please clone the course's [http://cartan.math.umb.edu/gitweb/?p=groebner.git;a=summary git repository] to your local machine and submit code assignments with <code>git push</code>.  (See [http://git-scm.com/book/en/Getting-Started-Installing-Git#Installing-on-Mac this link] for instructions on installing git, after which you might want to type <code>man gittutorial</code> from your machine's command line.)
 ;Course Textbook

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 Welcome to the wiki!  Editing this page is exactly like editing [http://en.wikipedia.org/wiki/Main_page Wikipedia].  You may wish to see their help pages on [http://en.wikipedia.org/wiki/Help:Contents/Editing_Wikipedia editing] and on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].
+; Code repository
+Please clone the course's [http://cartan.math.umb.edu/gitweb git repository] to your local machine and submit code assignments with <code>git push</code>.  (See [http://git-scm.com/book/en/Getting-Started-Installing-Git#Installing-on-Mac this link] for instructions on installing git, after which you might want to type <code>man gittutorial</code> from your machine's command line.)
 ;Course Textbook

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 == Course Structure ==
-A typical semester spent on this course will reach Chapter 7 of the course textbook, "Invariant Theory of Finite Groups". Progress through the material can either be measured weekly through the completion of programming assignments or short presentations of material. A source code repository for the course is available [http://code.google.com/p/umb-compuatational-algebraic-geometry/ here].
+A typical semester spent on this course will reach Chapter 7 of the course textbook, "Invariant Theory of Finite Groups". Progress through the material can either be measured weekly through the completion of programming assignments or short presentations of material. A source code repository for the course is available [http://code.google.com/p/umb-computational-algebraic-geometry/ here].
 * Chapter 1: Geometry, Algebra, and Algorithms

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 Or something similar; I don't think returning differently ordered lists as a class method is very Python like, which is why I'm leaning towards a library/module function. [[User:Patrickmclaren|Patrickmclaren]] 02:49, 3 May 2012 (BST)
+I think Sage has a class for rings and creates an instance of this class to store things like variable names and monomial orders.  Then it has enother class for ring elements; instances of this class store things like monomials and coefficients (and an identifier for the ring that the element belongs to).  Then the elements get simpler -- for example I think your polynomial above could get stored as a single dictionary, say
+<pre>p = {(2, 3, 1): 6, (4, 4, 2): 2}</pre>
+and the __repr__ method asks the ring for the variable names and the monomial ordering.
+[[User:Steven Glenn Jackson|Steven Glenn Jackson]] 12:04, 3 May 2012 (BST)

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 * Chapter 7: Invariant Theory of Finite Groups
 ** Programming Assignment:
+== Questions ==
+=== Programming Assignments ===
+==== Chapter 1 ====
+==== Chapter 2 ====
+===== Representing Multivariable Polynomials =====
+I think I'll implement the polynomials as a list of dictionaries, so what I'll be storing is a list of unordered monomials:
+Eg.
+<pre>p = [{ 'c' : 6, 'x_1' : 2 , 'x_2' : 3, 'x_3' : 1}, { 'c' : 2, 'x_1' : 4, 'x_2' : 4, 'x_3' : 2}]</pre>
+would represent <math>6x_1^2x_2^3x_3 + 2x_1^4x_2^4x_3^2</math>
+For a moment, I thought about implementing the polynomial as a class, and add orderings as a class method. However, I think it would be more correct to order a polynomial by calling a library function:
+<pre>ordered_p = polynomial.order(p, 'grevlex')</pre>
+Or something similar; I don't think returning differently ordered lists as a class method is very Python like, which is why I'm leaning towards a library/module function. [[User:Patrickmclaren|Patrickmclaren]] 02:49, 3 May 2012 (BST)

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 One may read about the history of Algebraic Geometry in the Wikipedia [http://en.wikipedia.org/wiki/Algebraic_geometry article] on the subject. There is also thorough examination of the history of the subject written by Dieudonne, which is available [http://mathdl.maa.org/images/upload_library/22/Ford/Dieudonne.pdf here].
-Algebraic Geometry is a very active research field, a quick glance at the recent additions to the [http://arxiv.org/list/math.AG/recent arXiv] supports this assertion. There are quite a few research opportunities for undergraduates interested in Algebraic Geometry, especially in the North-Eastern region of the US. [http://www.agneshome.org/ AGNES] is a series of biannual weekend workshops in algebraic geometry. MIT and Harvard [http://math.mit.edu/seminars/ags host] a Algebraic Geometry seminar. Oakland University in Michigan organizes a yearly conference in Algebraic Geometry ([https://sites.google.com/a/oakland.edu/algebra/home MCAG]). SIAM AG hosts a Algebraic Geometry [http://lists.siam.org/mailman/listinfo/siam-ag mailing list].
+Algebraic Geometry is a very active research field -- a quick glance at the recent additions to the [http://arxiv.org/list/math.AG/recent arXiv] supports this assertion. There are quite a few research opportunities for undergraduates interested in Algebraic Geometry, especially in the North-Eastern region of the US. [http://www.agneshome.org/ AGNES] is a series of biannual weekend workshops in algebraic geometry. MIT and Harvard [http://math.mit.edu/seminars/ags host] a Algebraic Geometry seminar. Oakland University in Michigan organizes a yearly conference in Algebraic Geometry ([https://sites.google.com/a/oakland.edu/algebra/home MCAG]). SIAM AG hosts a Algebraic Geometry [http://lists.siam.org/mailman/listinfo/siam-ag mailing list]. Finally, there is a REU program in Algebraic Geometry at [http://www.math.clemson.edu/~kevja/REU/ Clemson University] in South Carolina.
 == Course Structure ==

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 * Chapter 1: Geometry, Algebra, and Algorithms
 ** Programming Assignment:
-::Write a python program which can determine whether a polynomial is within the ideal generated by a given list of polynomials
+::Write a python program which can determine whether a polynomial is within the ideal generated by a given list of polynomials.
 * Chapter 2: Groebner Bases
 ** Programming Assignment:
+::Implement the multi-variable division algorithm in Python.
-* Chapter 3: Elimination Theor
+::Implement Buchberger's Algorithm.
+* Chapter 3: Elimination Theory
 ** Programming Assignment:
 * Chapter 4: The Algebra-Geometry Dictionary