ABSTRACT :

The talk is based on the eponymous article by
Richter-Gebert,  Bernd Sturmfels and  Thorsten Theobald

We  will try  to  give a  simple  and elementary  introduction to  the
subject.  

'Tropical geometry' is a emergent  topic which appeared to be relevant
in different  mathematical areas such as  dynamical systems, algebraic
geometry, symplectic geometry, combinatorics ...

Roughly speaking a tropical curve looks like a piecewise-linear graph;
Generically  it  is  a  trivalent  graph in  the  so  called  tropical
projective  space.   Surprisingly  enough,  such objects  (and  their
higher  dimensional counterparts) have  nice geometrical  properties :
For example the  usual Bezout theorem of algebraic  geometry 'holds' in
this context.

Just a little more formally, tropical hypersurfaces appear as the most
'natural'  object (so  far)  when one  wants  to do  geometry with  base
semi-ring the min-plus algebra (R,min,+) (i.e. in R^n considered
as a module over (R,min,+)). 

In this first  introductory talk we will state  a formal definition of
tropical  hypersurfaces and we'll  try, through  lots of  examples and
pictures,  to  give  a  good  intuition  of  the  behaviors  of  these
objects.  Then   we  will  discuss   how  to  state  a   valid  Bezout
theorem.  Finally we will  see some  unfriendly behaviors  of tropical
hypersurfaces.