In this letter we show that the results of degree growth (algebraic entropy)
calculations for lattice equations strongly depend on the initial value
problem that one chooses. We consider two problematic types of initial
value configurations, one with problems in the past light-cone, the other
one causing interference in the future light-cone, and apply them to Hirota’s
discrete KdV equation and to the discrete Liouville equation. Both of these
initial value problems lead to exponential degree growth for Hirota’s dKdV, the
quintessential integrable lattice equation. For the discrete Liouville equation,
though it is linearizable, one of the initial value problems yields exponential
degree growth whereas the other is shown to yield non-polynomial (though still
sub-exponential) growth. These results are in contrast to the common belief that
discrete integrable equations must have polynomial growth and that linearizable
equations necessarily have linear degree growth, regardless of the initial value
problem one imposes. Finally, as a possible remedy for one of the observed
anomalies, we also propose basing integrability tests that use growth criteria on
the degree growth of a single initial value instead of all the initial values.