Tropical sequences associated with Somos sequences

V. A. Bykovskii ; M. A. Romanov ; A. V. Ustinov

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 118-132.

Résumé

Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-$k$ sequence for $k=4,5,6,7$ was proved. In the works of Hone, representations for Somos-$4$ and $5$ sequences were found via the Weierstrass sigma function on elliptic curves, and for $k=6$ via the Klein sigma function on hyperelliptic curve of genus $2$. It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For $k=4,5,6,7$, the Somos sequences are Laurent polynomials of $k$ initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables. Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free terms are written explicitly.

Détail

Keywords: Somos sequences, tropical sequences.

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