Tropical projective space

In tropical geometry, a tropical projective space is the tropical analog of the classic projective space.

Conventional visualization of the tropical projective plane, with projection of the real coordinate axes.

Definition

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Given a module M over the tropical semiring T, its projectivization is the usual projective space of a module: the quotient space of the module (omitting the additive identity 0) under scalar multiplication, omitting multiplication by the scalar additive identity 0:[a]

 

In the tropical setting, tropical multiplication is classical addition, with unit real number 0 (not 1); tropical addition is minimum or maximum (depending on convention), with unit extended real number (not 0),[b] so it is clearer to write this using the extended real numbers, rather than the abstract algebraic units:

 

Just as in the classical case, the standard n-dimensional tropical projective space is defined as the quotient of the standard (n+1)-dimensional coordinate space by scalar multiplication, with all operations defined coordinate-wise:[1]

 

Tropical multiplication corresponds to classical addition, so tropical scalar multiplication by c corresponds to adding c to all coordinates. Thus two elements of   are identified if their coordinates differ by the same additive amount c:

 

Notes

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  1. ^ As usual, scalar multiplication of any vector by 0 yields the identity for vector addition 0, so these must be omitted or all vectors will be identified.
  2. ^ can be interpreted as either positive or negative infinity, depending on convention.

References

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  1. ^ Mikhalkin 2006, p. 6, example 3.10.
  • Richter-Gebert, Jürgen; Sturmfels, Bernd; Theobald, Thorsten (2003). "First steps in tropical geometry". arXiv:math/0306366.
  • Mikhalkin, Grigory (2006). "Tropical Geometry and its applications". arXiv:math/0601041.