Tropical projective space

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:^[a]

\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 $\infty$ (not 0),^[b] 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:^[1]

\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}).

Notes

^ 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.
^ $\infty$ can be interpreted as either positive or negative infinity, depending on convention.

References

^ 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.

[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] $\infty$ can be interpreted as either positive or negative infinity, depending on convention.

[FOOTNOTEMikhalkin20066example_3.10-3] Mikhalkin 2006, p. 6, example 3.10.

[a]

[b]

[1]