Tropical projective space explained

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

Definition

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

T (M) : = (M \setminus 0 )/( 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:

T (M) : = (M \setminus \boldsymbol{ inf ty})/( T \setminus inf ty).

Just as in the classical case, the standard -dimensional tropical projective space is defined as the quotient of the standard -dimensional coordinate space by scalar multiplication, with all operations defined coordinate-wise:

TP ^{n} : = ( T ^{n+1} \setminus \boldsymbol{ inf ty})/( T \setminus inf ty). Tropical multiplication corresponds to classical addition, so tropical scalar multiplication by corresponds to adding to all coordinates. Thus two elements of are identified if their coordinates differ by the same additive amount :(x _{0,} ... , x _{n)} \sim (y _{0,} ... , y _{n)} \iff (x _{0} + c, ... , x _{n} + c) = (y _{0,} ... , y _{n).}

References

• math/0306366 . First steps in tropical geometry. Richter-Gebert. Jürgen. Sturmfels. Bernd. Theobald. Thorsten. 2003.
• math/0601041 . Tropical Geometry and its applications. Mikhalkin. Grigory. 2006.