On algebraic relations between solutions of a generic Painleve equation
On algebraic relations between solutions of a generic Painleve equation
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关于一般 Painleve 方程解之间的代数关系
DOI：
10.1515/crelle-2014-0082
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
A. Pillay
中科院分区：
文献类型：
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作者：
Ronnie Nagloo;A. Pillay
证明了如果y”= f（y，y '，t，\alpha，\beta，.）是一个通用的Painleve方程（即一个方程在家庭PI-PVI之一，但与复杂的参数\alpha，\beta，..代数独立），则一组解与它们的导数（y_1，...，y_n，y_1'，..，y_n ′）由一对解及其导数（y_i，y_i ′，y_j，y_j ′）证明。该证明结合了日本学校的工作“不可约”的Painleve方程，与trichomoty定理的强极小集在差分封闭领域。
We prove that if y" = f(y,y',t,\alpha, \beta,..) is a generic Painleve equation (i.e. an equation in one of the families PI-PVI but with the complex parameters \alpha, \beta,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school on "irreducibility" of the Painleve equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.