Computing Tropical Varieties Over Fields with Valuation
Thomas Markwig, Yue Ying Ren · Foundations of Computational Mathematics · 2019
We show how the tropical variety of an ideal $$I\unlhd K[x_1,\ldots ,x_n]$$ over a field K with non-trivial discrete valuation can always be traced back to the tropical variety of an ideal $$\pi ^{-1}I\unlhd R\llbracket t\rrbracket [x_1,\ldots ,x_n]$$ over some dense subring R in its ring of integers. We show that this connection is compatible with the Gröbner polyhedra covering them. Combined with previous works, we thus obtain a framework for computing tropical varieties over general fields with valuations, which relies on the existing theory of standard bases if $$\pi ^{-1}I$$ is generated by elements in $$R[t,x_1,\ldots ,x_n]$$ .