Bertrand Eynard, Institut de Physique Th茅orique, CEA (Commissariat 脿 l'Energie Atomique et aux Energies Alternative)
Title: Integrable systems from a Newton polygon.
础产蝉迟谤补肠迟:听Let $P(x,y)in mathbb C[x,y]$ a bivariate polynomial, written $P(x,y)=sum_{(i,j)in N}P_{i,j} x^i y^j$. The polytope $Nsubset mathbb Z imes mathbb Z$ is called the Newton's polytope and its convex envelope is the Newton's polygon. The locus of solutions $P(x,y)=0$ in $mathbb C imes mathbb C$ is a plane curve (an immersed Riemann surface). Its geometry can be produced from the combinatorics of the Newton's polygon. This allows to construct a Baker-Akhiezer function and a Lax pair associated to $P(x,y)$, in other words an integrable system. In fact there are several integrable systems that can be associated to P. The easiest is the isospectral system, following the Krichever's reconstruction method, and the others can be seen as "quantum" deformations. In particular we can get an isomonodromic system and other deformations. We shall illustrate on the elliptic curve $P(x,y)=y^2-x^3+g_2 x+x_3$.
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