A new type of exact solvability is reported. Schrödinger equation is considered in a very large spatial dimension D ≫ 1 and its central polynomial potential is allowed to depend on "many" (= 2q) coupling constants. In a search for its bound states possessing an exact and elementary wave functions ψ (proportional to a harmonicoscillator-like polynomial of a freely varying, i.e., not just small, degree N), the "solvability conditions" are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of D, q and N. Assuming that D is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all N and up to q = 5 at least.
In this paper, effectiveness of involutive criteria in the elimination of useless prolongations when computing polynomial Janet bases, which are typical representatives of involutive bases, is discussed. One of the results of this study is that the role of the criteria in an involutive algorithm is not as important as in the Buchberger algorithm. It is shown also that these criteria affect the growth of intermediate coefficients.
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