Recent Talks

• Observations on Toric Surface Codes -- HC Faculty Seminar, September 8 and 15, 2011.
• Error Control Codes from Algebra -- SACNAS National Meeting, Dallas TX, October 16, 2009.
• Order domains, Algebra Seminar, University of South Alabama, April 6, 2007.
• Gr"obner bases and polynomial equations Mathematics Department Colloquium, University of South Alabama, April 5, 2007.
• Mathematics and Music -- MAA NES meeting Sacred Heart University, November 18, 2006.
• Workshop on coding theory at TAGS, May 17 - 19, 2006
• Slides for a talk on Gr"obner bases for encoding of AG Goppa codes and codes from order domains (Gr"obner Bases Semester, D1 Workshop, Johann Kepler Universitaet, Linz), May 1, 2006
• Slides for talk in AMS Special Session on Coding and Design-Theoretic Applications of Polynomials, January 7, 2004.

Recent Publications

• (with David Damiano), A Course In Linear Algebra, corrected reprint, Dover, 2011.

• (with David Cox and Hal Schenck) Toric Varieties, AMS Graduate Studies in Mathematics 124, 2011.

• Algebraic geometry codes from higher-dimensional varieties, in Advances in Algebraic Geometry Codes, E. Martinez-Moro, Carlos Munera, and D. Ruano, eds. World Scientific, 2009.

• Automorphisms and Encoding of AG and Order Domain Codes, in: Gr"obner Bases, Coding Theory, and Cryptography, M. Sala, T. Mora, L. Perret, S. Sakata, C. Traverso, eds. Springer RISC series, 2009.

• A key equation and the computation of error values for codes from order domains (a shorter version of this paper appears in ADVANCES IN CODING THEORY AND CRYPTOGRAPHY, edited by T Shaska, W C Huffman, D Joyner, and V Ustimenko, World Scientific, 2007.)

• The Ubiquity of Order Domains for the Construction of Error Control Codes, Advances in Mathematics of Communications 1 (2007), 151-171; Erratum for: The Ubiquity of Order Domains for the Construction of Error Control Codes, Advances in Mathematics of Communications 2 (2008), 344-345.

• (with D. Cox and D. O'Shea), Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3nd edition, New York: Springer-Verlag, 2007. Follow this link for the Ideals, Varieties, and Algorithms homepage.

• (with Hal Schenck) Toric surface codes and Minkowski sums, SIAM Journal of Discrete Mathematics 20 (2006), 999-1014.

• (with Ryan Schwarz) Toric codes and Vandermonde determinants (new title) Applicable Algebra in Engineering, Communications, and Computing 18 (2007), 349-367.

• (A Holy Cross Project with Cristina Ballantine, Sharon Frechette) Determinants Associated to Zeta Matrices of Posets, Linear Algebra and its Applications 411 (2005), 364-370.

• (with D. Cox and D. O'Shea), Using Algebraic Geometry, 2nd ed., Graduate Texts in Mathematics, v. 185. New York: Springer-Verlag, 2005. This version contains several new sections on Gr"obner fans of ideals, the Gr"obner Walk basis conversion algorithm, and a new chapter on codes from order domains and the Berlekamp-Massey-Sakata decoding algorithm.

• (with Leah Gold and Hal Schenck) Cayley-Bacharach and Evaluation Codes on Complete Intersections, Journal of Pure and Applied Algebra, 196 (2005), 91-99.

• (with G. Boros, V. Moll, E. Mosteig, R. Stanley) A map on the space of rational functions, Rocky Mountain Mathematical Journal 35 (2005), 1861-1880.

• On the Zeroes of Two Families of Polynomials Arising From Certain Rational Integrals, Rocky Mountain Mathematical Journal 35 (2005), 1205 - 1216. This note gives a proof of a conjecture of Boros, Moll and Shallit on the locations of roots of two families of univariate polynomials arising in the evaluation of certain rational integrals. This problem was suggested to me by Victor Moll in June 2002 at SIMU at the University of Puerto Rico in Humacao. See Victor's article in the March 2002 issue of the Notices of the AMS for more on the background of this story.

• "Solving the Selesnick-Burrus Filter Design Equations Using Computational Algebra and Algebraic Geometry", Advances in Applied Mathematics 31 (2003), 463-500.

• (with D. Ortiz, R. Ortiz-Rosado, R. Pablo, and K. Rios-Soto) Some Remarks on Fitzpatrick and Flynn's Gr"obner Basis Technique for Pade' Approximation, Journal of Symbolic Computation 35 (2003), 451-461.

• (with D. Cox and D. O'Shea), Using Algebraic Geometry, Graduate Texts in Mathematics, v. 185. New York: Springer-Verlag, 1998.

• Canonical Curves and the Petri Scheme, in Groebner Bases and Applications, Buchberger, B. and Winkler, F. editors, London Mathematical Society Lecture Note Series, v. 251. Cambridge University Press, 381-392.
• Applications to Coding Theory, in Applications of Computational Algebraic Geometry, AMS Proceedings of Symposia in Applied Mathematics, 53, 143--167 (from the AMS Short Course in Applications of Computational Algebraic Geometry, San Diego, CA, January 1997).
• (with C. Heegard and K. Saints), On the Structure of Hermitian Codes, Journal of Pure and Applied Algebra 121 (1997), 293--314.
• (with D. Cox and D. O'Shea), Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edition, New York: Springer-Verlag, 1996. Follow this link for the Ideals, Varieties, and Algorithms homepage.

• (with C. Heegard and K. Saints), Systematic Encoding via Groebner Bases for a Class of Algebraic Geometric Goppa Codes, IEEE Transactions on Information Theory, v. 41, no. 6 (November 1995), 1752--1761.
• The Algebraic Structure of Some AG Goppa Codes, in Proceedings of the 33rd Annual Allerton Conference on Communication, Control, and Computing, October 4-6, 1995.

• My homepage
• Math/CS homepage
• Holy Cross homepage

Last modified: September 15, 2011