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John McKay CHALLENGE system of polynomial equations

# This is a system of polynomials for the set of
# twelve q-coefficients {a1..5,a7..9,a11,a17,a19,a23}
# of replicable fns. Remove the solns ak=0 for all k > 0.
# It is an important problem to solve even in Z .
# There is a check at the end. Should give g1..g17 = 0.
# Apart from the solution a[k]= 0 for all k > 1,
# the rest should be a zero-dimensional ideal.
# There are about 600+ rational integer solutions.

Here is the system (Maple code)

Here is the polynomial j (Maple code)

Reference:
The Essentials of Monstrous Moonshine,
John McKay - Advanced Studies in Pure Mathematics 32, 2001,
Groups & Combinatorics - in memory of Michio Suzuki.

The AMS MathSciNet Review of this reference can be found
here (linked .pdf file)

Contact: John McKay, Concordia University,
**mckay@cs.concordia.ca**

John McKay gave a talk at the Perimeter Institute (Waterloo ON) in March 2007
see the poster here (linked .pdf file)