Yue Ren (Kaiserslautern)

02.06.2016 - W01 0-012 (Wechloy), 16 Uhr c.t.

Constructive tropical geometry over fields with valuation

Gröbner bases techniques have become a staple in computational
algebraic geometry. From the dimensions of algebraic varieties to
resolutions of isolated singularities, almost every algorithm relies
on them.

However, should the ground field carry a non-trivial valuation that is
relevant for the question at hand, these techniques seemingly reach
their limit, as the underlying monomial orderings ignore the
coefficients. This happens for example in tropical geometry, in which
we use a valuation on the ground field to cast a combinatorial shadow
of an algebraic variety, that we then study in its place.

For this reason, there exist a series of works which extend the
classical Gröbner basis theory to take valuations on the ground field
into account. But this approach requires a reimplementation of a
modified Buchberger algorithm. While both algorithms are of the same
complexity, current implementations of the classical Buchberger
algorithm are so heavily optimized that the lack thereof strongly
impedes the performance for practical applications.

In this talk, we will discuss an alternative approach to the problem of
working over fields with valuation with Gröbner bases techniques. By
introducing a new variable and falling back to the ring of integers,
we can emulate the valuation on the ground field to solve all
computational problems in tropical geometry using the tools that
already exist in many computer algebra programs. We will highlight
some computational challenges that arise on the way and show how
overcoming them leads to better algorithms for other applications as
well.