A Bare-Bones Chronology of Somos Sequences

When people ask me for the chronology of Somos sequences, what they usually have in mind is Who first proved integrality of this-or-that particular Somos sequence? This web-page is my attempt to answer that question. However, it is not clear to me that this is the most important question.

Integrality of Somos-4 and Somos-5 was proved independently by quite a few people in 1990 (such as Janice Malouf, Enrico Bombieri, and Dean Hickerson). Janice Malouf published her proof in 1992 (Janice L. Malouf, An integer sequence from a rational recursion, Discrete Mathematics 110, pp. 257-261).

Dean Hickerson proved integrality of Somos-6 in April 1990. Ben Lotto proved integrality of Somos-7 later that month, using the same method as Dean Hickerson. The method was highly computational and the work was never published.

All four of these integrality results (Somos-k for k = 3, 4, 5, and 6) were proved in a unified way by Sergey Fomin and Andrei Zelevinsky in 2001, as described in their article The Laurent phenomenon. A more combinatorial approach to Somos-4 and Somos-5 was found by David Speyer (see his article Perfect matchings and the octahedron recurrence), based in part on unpublished work of Jim Propp, and this was extended to Somos-6 and Somos-7 by David Speyer in collaboration with Gabriel Carroll (see their article The cube recurrence).

It should be stressed that the work of Fomin, Zelevinsky, Speyer, and Carroll deals with much more than just Somos sequences! Specifically, it appears that Somos-4 and Somos-5 are best understood as special cases of three-term Gale-Robinson recurrences, while Somos-6 and Somos-7 are best understood as special cases of four-term Gale-Robinson recurrences.

Mireille Bousquet-Mélou and Julian West (using the same unpublished work of Jim Propp mentioned above) have found an explicit combinatorial model for all three-term Gale-Robinson recurrences (Speyer's work implies that such models can be found, but his interest was in a much more general setting, and his paper does not treat Gale-Robinson recurrences specifically). This work is currently being written up for publication but is not yet available.

Integrality is not the only important issue for Somos sequences. For instance, in 1993 Somos found an astonishing exact formula for Somos-6. Similar formulas have been found for Somos-4 and Somos-5 (see e.g. the write-up on Somos-4 produced by Noam Elkies in conversation with other members of the sci.math.research discussion group) but as far as I know, nobody has done anything similarly explicit for Somos-7.

One very neglected issue is, what do Somos sequences really mean? My own belief is that the best answer will come from a combination of integrable systems theory and statistical mechanics. But much remains to be done, and there are still no clear links between the analytic and combinatorial sides of the subject. In many important respects, we still don't understand Somos sequences much better than we did in 1990.

Jim Propp, July 2006