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SUMMARY:Exact results for degree growth of lattice equations and their sig
 nature over finite fields - Roberts\, J (University of New South Wales)
DTSTART:20130711T083000Z
DTEND:20130711T090000Z
UID:TALK46182@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:In the first part of the talk\, we study the growth of degrees
  (algebraic entropy) of certain\nmulti-affine quad-rule lattice equations 
 with corner boundary conditions. We work projectively with a free paramete
 r in the boundary values\, so that at each vertex\, there are 2 polynomial
 s in this parameter. \nWe show the ambient growth of their degree is known
  exactly\, via the asymptotics of the\nDelannoy double sequence. Then we g
 ive a conjectured growth for the degrees of the greatest common divisor th
 at is cancelled at each vertex. Taken together\, these provide us with a c
 onstant coefficient linear partial difference equation that determines the
  growth in the reduced degrees at each vertex. For a whole class of equati
 ons\, including most of the ABS list\, this proves polynomial growth of de
 gree. For other equations where the cancellation at each vertex is not hig
 h enough\, we prove exponential growth.\nIn the second part of the talk\, 
 we study integrable lattice equations and their perturbations\nover finite
  fields. We discuss some tests that can distinguish between integrable equ
 ations and their non-integrable perturbations\, and their limitations. Som
 e integrable candidates found using these tests can then be shown to have 
 vanishing entropy via the results of the first part of the talk. \nBoth pa
 rts of the talk are joint work with Dinh Tran (UNSW).\n
LOCATION:Seminar Room 1\, Newton Institute
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