When a packing is found empirically, there is always
the question of whether it is really exists. Since the coordinates
as numbers are always slightly in error, possibly the circles overlap,
say by some tiny amount. Possibly the system of equations is inconsistent,
but just barely so. If by numerical methods, a solution of the equations
converges without difficulty we can be confident that a solution really
exists. We can be certain the packing exists if, in addition, the
polynomial equations can be solved by algebraic means, or at least reduced
to a univariate polynomial. Sometimes this can be done, when n is
sufficiently small, by Groebner basis methods or by using resultants with
maple or other computer algebra system.
Return to minimal polynomials page.