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Regul. Chaotic Dyn., 2017, Volume 22, Issue 4, Pages 386–497 (Mi rcd262)  

Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus

Thierry Combot

Scuola Normale Superiore, Centro di Ricerca Matematica Ennio De Giorgi, Laboratorio Fibonacci, Piazza Cavalieri, 56127 Pisa

Abstract: We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong rational integrability condition on $V$, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions $2$ and $3$ and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated.

Keywords: trigonometric polynomials, differential Galois theory, integrability, Toda lattice

DOI: https://doi.org/10.1134/S1560354717040049

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Bibliographic databases:

Document Type: Article
MSC: 37J30
Received: 27.04.2017
Accepted:01.06.2017
Language: English

Citation: Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497

Citation in format AMSBIB
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\by Thierry Combot
\paper Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 4
\pages 386--497
\mathnet{http://mi.mathnet.ru/rcd262}
\crossref{https://doi.org/10.1134/S1560354717040049}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85026864525}


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