Supertropical semirings and supervaluations
Abstract
We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so called bipotent semiring $M$ (the usual maxplus setting), and then define a \textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U \to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $v$ a Krull valuation, we give a complete explicit description of $\Cov(v)$. The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual maxplus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.1101
 Bibcode:
 2010arXiv1003.1101I
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 13A18;
 13F30;
 16W60;
 16Y60
 EPrint:
 47 pages