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Tropical ideals. (English) Zbl 1428.14093
Summary: 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 J. Giansiracusa and N. Giansiracusa [Duke Math. J. 165, No. 18, 3379–3433 (2016; Zbl 1409.14100)]. 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.

14T10 Foundations of tropical geometry and relations with algebra
05B35 Combinatorial aspects of matroids and geometric lattices
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