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Mathematics Conferences Mini-workshop on Algebraic geometry and its Applications
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"Degree Complexity: How difficult is it to compute Groebner bases?"
SPEAKER  |  Ahn, Jeaman
INSTITUTE  |  Kongju National University
DATE  |  February 21(Fri), 2014
TIME  |  14:00
PLACE  |  Room 403, Center for Mathematical Challenges (CMC)
Keyword  |  
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ABSTRACT  |  How difficult is it to compute Gr?"{o}bner Bases? Since the construction of Gr?"{o}bner bases provides the foundation for most computation in algebraic geometry and commutative algebra, it is important to know the complexity of the computation of Gr?"{o}bner Bases. For a homogeneous ideal I in a polynomial ring R=k[x0,,xn], Bayer and Mumford introduced the regularity of the initial ideal of I with respect to a term order as a measure of the complexity of computing Gr?bner bases. One of the important problems is to bound the regularity of an initial ideal of I for a given term order on monomials. Two typical orderings are the graded lexicographic ordering and the graded reverse lexicographic order. Many people have studied this problem with respect to the reverse lexicographic term ordering, as these ideals have essentially best-case complexity due to a result of Bayer and Stillman. However, much less is known about regularities of initial ideals with respect to the lexicographic term ordering. One expects them to require many more generators than the reverse lexicographic initial ideals, but their precise behavior has been very little known. In this talk, I will give a survey of the recent progress on this topic and introduce a relationship between the complexity of algebraic computations with the ideal of a smooth variety and the geometry of its generic projection.
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