Abstract. Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the so(3) Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd’s Diophantine integrability criterion.

Rough notes in preparation for a lecture at the joint AMS-MAA conference, Jan. 5, 2012 We mathematicians have handy ways of discovering what stands a chance of being true. And we have a range of different modes of evidence that help us form these expectations; such as: analogies with things that are indeed true, computations, special case justifications, etc. They abound, these methods—explicitly formulated, or not. They lead us, sometimes, to a mere hint of a possibility that a mathematical statement might be plausible. They lead us, other times, to substantially firm—even though not yet justified—belief. They may lead us astray. Our end-game, of course, is understanding, verification, clarification, and most certainly: proof; truth, in short. Consider the beginning game, though. With the word “plausible ” in my title, you can guess that I’m a fan of George Pólya’s classic Mathematics and Plausible Reasoning ([?]). George Pólya (1887-1985) I think that it is an important work for many reasons, but mainly because Pólya is pointing to an activity that surely takes up the majority of time, and energy, of anyone engaged in thinking 1 about mathematics, or in trying to work towards a new piece of mathematics. Usually under limited knowledge and much ignorance, often plagued by mistakes and misconceptions, we wrestle

Abstract. Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the so(3) Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd’s Diophantine integrability criterion. 1.