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On the algebraic independence of generic Painlevé transcendents

  • Joel Nagloo (a1) and Anand Pillay (a2)

Abstract

We prove that if $y''=f(y,y',t,\alpha ,\beta ,\ldots)$ is a generic Painlevé equation from among the classes II, IV and V, and if $y_1,\ldots,y_n$ are distinct solutions, then $\mathrm{tr.deg}(\mathbb{C}(t)(y_1,y'_1,\ldots,y_n,y'_n)/\mathbb{C}(t))=2n$ . (This was proved by Nishioka for the single equation $P_{{\rm I}}$ .) For generic Painlevé III and VI, we have a slightly weaker result: $\omega $ -categoricity (in the sense of model theory) of the solution space, as described below. The results confirm old beliefs about the Painlevé transcendents.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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