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基本説明
The 16th Problem of Hilbert is one of the most famous remaining unsolved problems of mathematics. It concerns whether a polynomial vector field on the plane has a finite number of limit cycles.
Full Description
The 16th Problem of Hilbert is one of the most famous remaining unsolved problems of mathematics. It concerns whether a polynomial vector field on the plane has a finite number of limit cycles. There is a strong connection with divergent solutions of differential equations, where a central role is played by the Stokes Phenomenon, the change in asymptotic behaviour of the solutions in different sectors of the complex plane.The contributions to these proceedings survey both of these themes, including historical and modern theoretical points of view. Topics covered include the Riemann-Hilbert problem, Painleve equations, nonlinear Stokes phenomena, and the inverse Galois problem.
Contents
Non-accumulation of limit cycles - revisiting and simplyifyng a former proof; followed by construction of a summit-crossing "central trajectory" at semihyperbolic points, J. Ecalle; finiteness theorems for limit cycles - functional cochains, bifurcations and zeros of Abelian integrals, Y. Il'yashenko; introduction to Hilbert's 16th problem, J.-P. Ramis; isoresurgent deformations, J. Ecalle; isomonodromy deformations and connection formulae for Painleve transcendents, A.R. Its; monodromy groups of regular systems on CP1 and their invariants, V. Kostov; Galois groups for difference equations, M. van der Put; a differential analog of Abhyabkar's conjecture - some analogies between Stokes phenomena and wild ramification, J.-P. Ramis; Stokes phenomena in two dimensions, C. Sabbah; the inverse Galois problem for differential equations, M. Singer; algorithmic approach of the multisummation of formal power series solutions of linear ODE, applied to the Stokes phenomena, J. Thomann; and other papers.
