Gröbner bases over fields with valuation and tropical curves by coordinate projections
Andrew J. Chan · Warwick Research Archive Portal (University of Warwick) · 2013
In the emerging field of tropical geometry, algebraic varieties are replaced by polyhedral objects called tropical varieties. The algebraic and tropical variety share many invariants, but due to its polyhedral structure the tropical variety is often easier to work with. In this thesis, we look at two problems related to constructing tropical varieties. In the first, we extend the theory of Grobner bases to the case where we are looking over a field with a valuation. The motivation is that we can use these Grobner bases in order to compute tropical varieties over fields with valuations. We discuss some complexity and implementation issues and present a family of ideals whose Grobner basis with respect to the p-adic valuation is small, but all of whose standard Grobner bases are large. In the second, we investigate finding tropical curves over fields with the trivial valuation from their two-dimensional coordinate projections. A tropical curve has the support of a one-dimensional fan, and we use its coordinate projections to reconstruct the rays of this fan. We discuss some implementation issues and we see examples of tropical curves which can be computed using our projection techniques which cannot be computed with existing techniques.