Recent Talks
- 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.
- Toric codes Discrete Math Day at Holy Cross,
November 11, 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
- Automorphisms and Encoding of AG
and Order Domain Codes, to appear in volume from D1 Workshop
on applications of Gr"obner bases in coding theory and cryptography,
RISC-Linz.
- 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.
- (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 in version to be published)
to appear Applicable Algebra in Engineering,
Communications, and Computing.
- (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.
Last modified: August 16, 2007