Computing of tropical sequences associated with Somos sequences in the Gfan package
DOI:
https://doi.org/10.32603/2071-2340-2024-1-18-31Keywords:
tropical semiring, tropicalization, tropical prevariety, tropical recurrent sequence, tropical entropy, gfan packageAbstract
This paper examines tropical recurrent sequences associated with Somos sequences. The classical Somos sequences have applications in the theory of elliptic curves. Due to the Laurent character of the classical sequences, a pattern can be inferred between the classical sequences and their tropical counterparts.
The greatest interest is the increase in the dimension of the solution space of tropical sequences depending on the length of the final sequence. For a set of tropical sequences described by tropical recurrence relations, D.Yu.Grigoriev put forward a hypothesis about the stabilization of the maximum dimensions of the components of the corresponding tropical prevarieties. This hypothesis has been proven for tropical linear recurrent sequences. As part of this work, for tropical recurrent sequences associated with the sequences Somos-4 and Somos-5, the corresponding tropical prevarieties were investigated using the Gfan package in order to test Grigoriev's hypothesis.
References
B. Sturmfels, Algebraic statistics for Computational Biology, Cambridge, England: Cambridge University Press, 2005.
E. A. Baldwin and P. D. Klemperer, Tropical Geometry to Analyse Damand, London: Grantham Research Institute, 2014.
L. Zhang, G. Naitzat, and L. Lim, “Tropical Geometry of Deep Neural Networks,” in Proc. of the 35th International Conference on Machine Learnin, vol. 80, pp. 5824–5832, 2018.
D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Providence, USA: American mathematical Society, 2015.
F. Mikhailov, “Computing of the Dimensions of the Components of Tropical Prevarieties Described by Linear Tropical Recurrent Relations,” Computer tools in education, no. 1, pp. 40–54, 2023 (in Russian).
N. Elizarov and D. Grigoriev, “A tropical version of Hilbert polynomial (in dimension one),” in arXiv, 2022. [Online]. Available: https://arxiv.org/abs/2111.14742
A. N. Jensen, “Algorithmic Aspects of Gr¨obner Fans and Tropical Varieties,” Ph.D. Theses, Department of Mathematical Sciences, University of Aarhus, Denmark, 2007.
C. S. Swart and A. N. W Hone, “Integrality and the Laurent phenomenon for Somos 4 sequences,” in arXiv , 2005. [Online]. Available: https://arxiv.org/abs/math/0508094
A. N. W. Hone, “Sigma function solution of the initial value problem for Somos 5 sequences,” in arXiv, 2005. [Online]. Available: https://arxiv.org/abs/math/0501554
Yu. N. Fedorov and A. N. W. Hone, “Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties,” in arXiv, 2015. [Online]. Available: https://arxiv.org/abs/1512.00056
S. Fomin and A. Zelevinsky, “The Laurent Phenomenon,” Advances in Applied Mathematics,” vol. 28, no. 2, pp. 119–144, 2002; doi:10.1006/aama.2001.0770.
D. Grigoriev, “Tropical recurrent sequences,” Advances in Applied Mathematics, vol. 116, p. 102012, 2020; doi:10.1016/j.aam.2020.102012
A. N. Jensen, Gfan version 0.6: A User’s Manual, Department of Mathematical Science, University of Aarhus, Denmark, 2017.
V. A. Bykovskii, M. A. Romanov, and A. V. Ustinov, “GTropical sequences associated with Somos sequences,” Chebyshevskii sbornik, vol. 22, no. 1, pp. 118–132, 2021 (in Russian); doi:10.22405/2226-8383-2021-22-1-118-132
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