Rational maps, Laurent polynomials, and combinatorics

Boston University Dynamics Seminar
January 31, 2005

James Propp
University of Wisconsin
(URL: jamespropp.org)


Certain rational maps from complex projective space to itself are governed by the still-mysterious "Laurent phenomenon" studied by Fomin and Zelevinsky. In many cases the Laurent polynomials that arise have been empirically observed to have only positive coefficients; in some of these cases, the only known proofs of positivity are combinatorial, and the proofs provide enumerative interpretations of the coefficients via partition functions of lattice models (such as dimer models on planar graphs).

For instance, the birational map (w:x:y) &rarr (wx:xy:w2+y2) from CP2 to itself is related to the dimer model on a 2-by-2n grid (which as n goes to infinity gives the golden-mean shift); the birational map (w:x:y:z) &rarr (wx:xy:xz:y2+z2) is related to the dimer model on bent versions of these graphs that have implications for the study of Markoff numbers; the birational map (v:w:x:y:z) &rarr (vw:wx:wy:wz:xz+y2) (associated with an integer sequence introduced by Michael Somos) is related to the dimer model on some strange-looking planar graphs; and other maps take us beyond dimer models altogether.

Many of these rational maps are best viewed as degenerations of algebraic cellular automata, in which spacetime is a lattice or a tree.

The talk will focus on concrete examples, since a general theory is still lacking. The talk will also include a discussion of the notion of algebraic entropy in the sense of Bellon and Viallet, and their intriguing still-open conjecture that it is always the logarithm of an algebraic integer.