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Tropical ideals

Published online by Cambridge University Press:  30 January 2018

Diane Maclagan  and
Felipe Rincón
Diane Maclagan
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email D.Maclagan@warwick.ac.uk
Felipe Rincón
Affiliation:
Department of Mathematics, University of Oslo, 0851 Oslo, Norway email feliperi@math.uio.no
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Abstract

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 3 , March 2018 , pp. 640 - 670
Copyright
© The Authors 2018 

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