Exact results for degree growth of lattice equations and their signature over finite fields
Seminar Room 1, Newton Institute
AbstractIn the first part of the talk, we study the growth of degrees (algebraic entropy) of certain multi-affine quad-rule lattice equations with corner boundary conditions. We work projectively with a free parameter in the boundary values, so that at each vertex, there are 2 polynomials in this parameter. We show the ambient growth of their degree is known exactly, via the asymptotics of the Delannoy double sequence. Then we give a conjectured growth for the degrees of the greatest common divisor that is cancelled at each vertex. Taken together, these provide us with a constant coefficient linear partial difference equation that determines the growth in the reduced degrees at each vertex. For a whole class of equations, including most of the ABS list, this proves polynomial growth of degree. For other equations where the cancellation at each vertex is not high enough, we prove exponential growth. In the second part of the talk, we study integrable lattice equations and their perturbations over finite fields. We discuss some tests that can distinguish between integrable equations and their non-integrable perturbations, and their limitations. Some integrable candidates found using these tests can then be shown to have vanishing entropy via the results of the first part of the talk. Both parts of the talk are joint work with Dinh Tran (UNSW).
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