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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Exact results for degree growth of lattice equatio
ns and their signature over finite fields - Robert
s\, J (University of New South Wales)
DTSTART;TZID=Europe/London:20130711T093000
DTEND;TZID=Europe/London:20130711T100000
UID:TALK46182AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/46182
DESCRIPTION:In the first part of the talk\, we study the growt
h of degrees (algebraic entropy) of certain\nmulti
-affine quad-rule lattice equations with corner bo
undary conditions. We work projectively with a fre
e parameter in the boundary values\, so that at ea
ch vertex\, there are 2 polynomials in this parame
ter. \nWe show the ambient growth of their degree
is known exactly\, via the asymptotics of the\nDel
annoy double sequence. Then we give a conjectured
growth for the degrees of the greatest common divi
sor that is cancelled at each vertex. Taken togeth
er\, 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 o
f the ABS list\, this proves polynomial growth of
degree. For other equations where the cancellation
at each vertex is not high enough\, we prove expo
nential growth.\nIn the second part of the talk\,
we study integrable lattice equations and their pe
rturbations\nover finite fields. We discuss some t
ests that can distinguish between integrable equat
ions and their non-integrable perturbations\, and
their limitations. Some integrable candidates foun
d using these tests can then be shown to have vani
shing entropy via the results of the first part of
the talk. \nBoth parts of the talk are joint work
with Dinh Tran (UNSW).\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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