Tropical projective space

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

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

Definition

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:

[math]\displaystyle{ \mathbf{T}(M) := (M \setminus \mathbf{0})/(\mathbf{T} \setminus 0). }[/math]

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), so it is clearer to write this using the extended real numbers, rather than the abstract algebraic units:

[math]\displaystyle{ \mathbf{T}(M) := (M \setminus \boldsymbol{\infty})/(\mathbf{T} \setminus \infty). }[/math]

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:

[math]\displaystyle{ \mathbf{TP}^n := (\mathbf{T}^{n+1} \setminus \boldsymbol{\infty})/(\mathbf{T} \setminus \infty). }[/math]

Tropical multiplication corresponds to classical addition, so tropical scalar multiplication by c corresponds to adding c to all coordinates. Thus two elements of [math]\displaystyle{ \mathbf T^{n+1} \setminus \boldsymbol{\infty} }[/math] are identified if their coordinates differ by the same additive amount c:

[math]\displaystyle{ (x_0, \dots, x_n) \sim (y_0, \dots, y_n) \iff (x_0 + c, \dots, x_n + c) = (y_0, \dots, y_n). }[/math]

References

• 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.