TL;DR
This paper proves the existence of generic tropical varieties for graded ideals with constant coefficients, paralleling the concept of generic initial ideals, and characterizes them for specific classes like principal and linear ideals.
Contribution
It establishes the existence of generic tropical varieties in the constant coefficient case and describes their structure for principal and linear ideals.
Findings
Generic tropical variety exists for graded ideals with constant coefficients.
The structure of the generic tropical variety is characterized as a set and as a fan for principal and linear ideals.
Analogous to the existence of generic initial ideals in Groebner basis theory.
Abstract
We show that in the constant coefficient case the generic tropical variety of a graded ideal exists. This can be seen as the analogon to the existence of the generic initial ideal in Groebner basis theory. We determine the generic tropical variety as a set in general and as a fan for principal ideals and linear ideals.
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