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:

$\mathbf {T} (M):=(M\setminus \mathbf {0} )/(\mathbf {T} \setminus 0).$

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:

$\mathbf {T} (M):=(M\setminus {\boldsymbol {\infty }})/(\mathbf {T} \setminus \infty ).$

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:

$\mathbf {TP} ^{n}:=(\mathbf {T} ^{n+1}\setminus {\boldsymbol {\infty }})/(\mathbf {T} \setminus \infty ).$

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

$(x_{0},\dots ,x_{n})\sim (y_{0},\dots ,y_{n})\iff (x_{0}+c,\dots ,x_{n}+c)=(y_{0},\dots ,y_{n}).$

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.