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On basic concepts of tropical geometry. (English. Russian original) Zbl 1237.14074

Proc. Steklov Inst. Math. 273, 252-282 (2011); translation from Tr. Mat. Inst. Steklova 273, 271-303 (2011).
When introducing tropical geometry to a newcomer, a common question is: “Why is it impossible to define a tropical hypersurface by the equation \(f=0\), where \(f\) is a tropical polynomial and \(0\) is the tropical zero (i.e., \(-\infty\)), instead of taking the corner locus of \(f\), which is so different from conventional algebraic geometry?” The easy answer is “because \(f=0\) has no solutions”, but the question itself is not silly, and points out an important gap in the common way to establish a framework for tropical geometry (see, for example, [B. Sturmfels, CBMS Regional Conference Series in Mathematics 97. Providence, RI: AMS (2002; Zbl 1101.13040)] or [G. Mikhalkin, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: EMS. 827–852 (2006; Zbl 1103.14034)] for commonly accepted introductions to the subject).
The author suggests an unexpectedly elementary and fruitful way to fix this problem: instead of defining the tropical sum of \(a\) and \(b\) as \(\max(a,b)\), one should preserve this definition for \(a\neq b\), and define the tropical sum of \(a\) and \(a\) as \([ -\infty, a]\), in the same way as it happens for the degree of the sum of two polynomials \(F\) and \(G\): we have \(\deg(F+G) = \max (\deg F, \deg G)\) provided that \(\deg F \neq \deg G\), otherwise \(\deg(F+G)\) may be equal to any number less than or equal to \(\deg F\). With this convention, a tropical hypersurface can be defined as the set of solutions of a tropical algebraic equation.
Besides this toy application, the idea of defining tropical operations as multivalued is used to introduce tropical operations on the set of complex numbers, resulting in the so-called complex tropical hyperfield. Complex tropical varieties, which appeared in many works in the role of crucial, but purely technical constructions (see, for example, complex tropical curves in [G. Mikhalkin, J. Am. Math. Soc. 18, 313–377 (2005; Zbl 1092.14068)]), turn out to be fully legitimate algebraic varieties over the complex tropical hyperfield, which allows further categorification of the theory. The author also suggests a framework to study topology of (multivalued) polynomials over complex tropical numbers.

MSC:

14T05 Tropical geometry (MSC2010)
12K10 Semifields
20N20 Hypergroups
54A99 Generalities in topology
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