  A new type of exact solvability is reported. We study the general central
polynomial potentials (with 2q anharmonic terms) which satisfy the Magyari's
partial exact solvability conditions (this means that they possess a
harmonic-oscillator-like wave function proportional to a polynomial of any
integer degree N). Working in the space of a very large dimension D for
simplicity, we reveal that in contrast to the usual version of the model in
finite dimensions (requiring a purely numerical treatment of the Magyari's
constraints), our large D problem acquires an explicit, closed form solution at
all N and up to q = 5 at least. This means that our effective secular
polynomials (generated via the standard technique of Groebner bases) happen to
be all fully factorizable in an utterly mysterious manner (mostly, over
integers).
