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2013
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# Thierry Combot

6 avenue Alain Savary 21000 Dijon
University of Burgundy

## Publications:

 Combot T. Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus 2017, vol. 22, no. 4, pp.  386-407 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 dimension 2 and 3, and recover several integrable cases. These potentials after a complex variable change 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 Citation: Combot T.,  Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 386-407 DOI:10.1134/S1560354717040049
 Combot T. Non-integrability of a Self-gravitating Riemann Liquid Ellipsoid 2013, vol. 18, no. 5, pp.  497-507 Abstract We consider the motion of a triaxial Riemann ellipsoid of a homogeneous liquid without angular momentum. We prove that it does not admit an additional first integral which is meromorphic in position, impulsions, and elliptic integrals which appear in the potential. This proves that the system is not integrable in the Liouville sense; we actually show that even its restriction to a fixed energy hypersurface is not integrable. Keywords: Morales–Ramis theory, elliptic functions, monodromy, differential Galois theory, Riemann surfaces Citation: Combot T.,  Non-integrability of a Self-gravitating Riemann Liquid Ellipsoid, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 497-507 DOI:10.1134/S1560354713050031