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# VIGRE seminar, fall 2001: Gröbner Basis Theory and
Applications

- Instructor
- Elizabeth Arnold
- Students enrolled
- Erik Baumgarten, Kelli Carlson, Amy Collins, Cody Patterson
(undergraduate mathematics students); Woonjung Choi, Teresa
Guagliardo, Robert Main (graduate mathematics students)
- Description
- Gröbner basis theory and the broader field of
Computational Commutative Algebra and Algebraic Geometry are young,
fast-growing, exciting fields of research. The applications of
Gröbner bases are wide and varying. These include, but are not
limited to: Commutative algebra and algebraic geometry, computer
science, engineering, statistics, graph theory, combinatorics,
robotics, wavelets, partial differential equations, automated
theorem proving, etc. We explored some elementary applications of
Gröbner bases including Elimination, Polynomial maps,
Varieties, Field extensions, Graph coloring and Integer
programming. Students did computations with a computer algebra
system such as Maple or CoCoA. The first part of the seminar
involved lectures on the basics of Gröbner basis theory and
included a guest lecture by the founder of Gröbner basis
theory and inventor of Buchberger's algorithm for computing
Gröbner bases, Bruno Buchberger, from the Research Institute
for Symbolic Computation in Linz, Austria. The second part involved
presentations by the students on different applications of
Gröbner bases. The third part consisted of lectures on more
in-depth aspects of Gröbner basis theory, with written
assignments requiring some research into the current literature. At
the end of the seminar students presented their work on these
problems.