Far Eastern Mathematical Journal

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Polynomial Somos sequences II

Romanov M. A.

2022, issue 1, P. 91-99
DOI: https://doi.org/10.47910/FEMJ202209

It was proved in [1] that for $k=4,5,6,7$ the elements of the Somos-$k$ sequence defined by the recurrence
$$S_k(n+k)S_k(n)=\sum_{1\leqslant i\leqslant k/2}\alpha_i x_0\dots x_{k-1}S_k(n+k-i)S_k(n+i)$$
and initial values $S_k(j)=x_j$ ($j=0,\dots,k-1$) are polynomials in the variables $x_0,\dots,x_{k-1}$. The unit powers of the variables $x_j$ in the factors \linebreak $\alpha_i x_0\dots x_{k-1}$ can be reduced. In this paper, we find the smallest values of these powers, at which the polynomiality of the above sequence is preserved.

Somos sequences, ultradiscrete sequences

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[1] V. A. Bykovskii, M. A. Romanov, “Polinomial'nye posledovatel'nosti Somosa”, Funkts. analiz i ego pril., 55:1 (2021), 20–32.
[2] S. Fomin and A. Zelevinsky, “The Laurent Phenomenon”, Adv. Appl. Math., 28 (2002), 119–144.
[3] R. Robinson, “Periodicity of Somos sequences”, Proceedings of the AMS, 116:3 (1992), 613–619.
[4] Allan P. Fordy and Andrew Hone, “Symplectic Maps from Cluster Algebras”, Symmetry, Integrability and Geometry: Methods and Applications, 7 (2011), 091, 12 pp.
[5] Yoichi Nakata, “The solution to the initial value problem for the ultradiscrete Somos-4 and 5 equations”, 2017, 13 pp., arXiv: 1701.04262.

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