Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Term Paper

[The following notes are about Steven Kleiman’s 2010 Undergraduate Seminar in Computational Commutative Algebra and Algebraic Geometry, in which the students give the lectures and write a term paper. The main page for this course is here.]

The term paper is to be a ten-page essay on a topic related to the course. The goal is for you to learn something new, and to explain it clearly to others in the class, or better, to other upper-class math majors. The paper must be written in a professional style, and formatted in AMS-LaTeX, like the papers in MIT’s Undergraduate Journal of Mathematics; please click here for some helpful resources. If you do a good job on your paper, then, possibly after further editing, it can be published in the next volume.

The Journal is stored in the following two MIT collections:

  • Hayden Library – Science Journals | QA.M679
  • Institute Archives – Noncirculating Collection 1 | QA1.M585

The following papers were written for 18.704, and may serve as models for yours:

  • Volume 1, 1999
    • Paul Grayson, Robotic Motion Planning, pp. 57-67.
  • Volume 2, 2000
    • Ted Allison, Complexity of Computations of Ideal Membership, pp. 1-9.
  • Volume 3, 2001
    • Ethan Cotterill, Syzygies over polynomial rings, pp. 29-41.
    • Geoffrey L. Goodell, Algebraic Coding Theory, pp. 71-80.
    • Matt Menke, Running time of Groebner Basis Algorithms, pp. 145-151.
    • Brian D. Smithling, A Proof of Hilbert’s Syzygy Theorem, pp. 199-207.
  • Volume 4, 2002
    • Peter Ahumada, An Algorithm for Integer Programming Problems, pp. 1-11.
    • Nicholas Cohen, Automatic Geometric-Theorem Proving, pp. 29-38.
    • Leah Schmelzer, Implicitization via Resultants, pp. 179-188.
  • Volume 5, 2003
    • Eric Schwerdtfeger, An Introduction to Symmetric Polynomials, pp. 265-273.
  • Volume 6, 2004
    • Paul Gorbow, Ideals from Graphs, pp. 69-84.
  • Volume 9, 2007
    • Hyeyoun Chung, Computing Invariants of Finite Groups, pp. 11-29.
    • Anand Deopurkar, Normalization of Algebraic Varieties, pp. 43-63.
    • Pablo Solis, Splines on a Finer Subdivision, pp. 133-142.
  • Volume 10, 2008
    • Alessandro Chiesa, Companion Matrices for Systems of Polynomial Equations, pp. 31-41.
    • Philip Engel, Colorings and Cycles of Graphs, pp.43-52
  • Volume 11, 2009
    • Qingchun Ren, Finding Generators for Invariant Rings, pp. 79-89.
    • Pablo Spivakovsky-Gonzalez, Primality Testing for Polynomial Ideals, pp.107-116

Our text, “Ideals, Varieties, and Algorithms,” describes a number of possible topics in Appendix D. More possibilities are found in the following books, which are on reserve in the Hayden Library Reserve Stacks.

  1. QA251.3.A32 1994
    Adams, W. and Loustaunau, P., “An introduction to Gr”obner bases,” American Mathematical Society, 1994.
  2. QA564.C6883.
    Cox, D., Little, J., and O’Shea, D., “Using Algebraic Geometry,” Graduate Texts in Math., 185. Springer-Verlag, 1998; second edition, 2005. Google books has most of the book online; you can find it HERE.
  3. QA1.S981 v.53.
    Cox D., and Sturmfels, B., “Applications of computational algebraic geometry, Lectures presented at the AMS Short Course held in San Diego, CA, January 6-7, 1997,” Proceedings Symposia Applied Math, 53, AMS Short Course Lecture Notes, Amer. Math. Soc., 1998.
  4. QA218.S65 2005
    Dickenstein, A., and Emiris, I., “Solving polynomial equations: foundations, algorithms, and applications,” Springer-Verlag, 2005.
  5. QA251.3.E38 1995
    Eisenbud, D., “Commutative algebra with a view toward algebraic geometry,” Springer-Verlag, 1995.
  6. QA251.3.G745 2002
    Greuel, G.-M., and Pfister, G., “A Singular introduction to commutative algebra,” Springer, 2002.
  7. QA564.S29 (2003).
    Schenck, H., “Computational Algebraic Geometry.” London Math. Soc. Student Texts, 58. Cambridge Univ. Press, 2003.
  8. QA1.R336 no.97.
    Sturmfels, B., “Solving Systems of Polynomial Equations,” CBMS Conference no. 97 (2002: Texas A & M University), Amer. Math. Soc., 2002.
  9. QA251.3.V365 (1998).
    Vasconcelos, W., “Computational Methods in Commutative Algebra and Algebraic Geometry,” with chapters by D. Eisenbud, D. Grayson, J. Herzog, and M. Stillman, Algorithms and Computation in Mathematics, 2. Springer-Verlag, 1998.
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