Géométrie Tropicale - ISISS M. Casagrande (Pieve di Soligo)

ISISS M. Casagrande (Pieve di Soligo)
On définie deux nouvelles opérations: a +b = min{a; b} et a x b = a + b,par exampl: 3 + 7 = 3, 3 x 7 = 10. Quelles sont les propriétés de cette nouvelle addition et multiplication? Est-ce qu'on peut "tout-faire" comme
avec l'addition et la multiplication que l'on connait? Peut-on défnir la soustraction et la division?

The first step is to retrace the construction of the structure (ℂ,+, x), starting from the Peano axioms, in order to obtain the tropical semiring (ℝU {+∞}, +, x).
As ℂ, such semiring is algebraically closed, that is every univariate polynomial can be factorized in linear factors, hence a tropical version of the fundamental theorem of algebra holds.
Finally, we study how tropical polynomials can give raise to tropical algebraic curves (in ℝxℝ in particular). It is difficult here to recognise the correspondence with classical geometry, that is lines, circumferences, parabolas . . . , as the curve essentially depends on the number of monomials rather than the degree of the polynomial.