05586493

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05586493 Inoue, Rei Takenawa, Tomoyuki A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice. Commun. Math. Phys. 289, No. 3, 995-1021 (2009). 2009 Springer-Verlag, Berlin EN tropical geometry cellular automata Theta functions Zbl 0281.30013

doi:10.1007/s00220-009-0815-3

Fay's trisecant identity for Riemann's theta function, associated with an algebraic curve, was established in [{\it J. D. Fay}, Theta functions on Riemann surfaces. Lecture Notes in Mathematics. 352. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0281.30013)]. In the paper under review, the tropical version of this identity is established for certain special hyperelliptic tropical curves (and conjectured for an arbitrary tropical curve). In particular, the ultra-discrete periodic Toda lattice system $T$ can be transformed into a special case of Fay's trisecant identity for a tropical spectral curve of $T$, which allows the authors to obtain the general solution of $T$ in terms of the corresponding tropical Riemann's theta function. Alexander Esterov (Moscow)