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 \(R[t,x_1,\ldots ,x_n]\).

中文翻译：

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通过评估计算田间的热带品种

我们展示了如何很好地将理想\（I \ unlhd K [x_1，\ ldots，x_n] \）在具有非平凡离散估值的字段K上的热带变种追溯到理想\（\ pi ^ {-1} I \ unlhd R \ llbracket t \ rrbracket [x_1，\ ldots，x_n] \）在它的整数环中的某个稠密子环R上。我们证明此连接与覆盖它们的Gröbner多面体兼容。与先前的工作相结合，我们因此获得了一个用于计算具有估值的一般领域中热带品种的框架，如果\（\ pi ^ {-1} I \）由\（R [t，x_1，\ ldots，x_n] \）。