Extend Moh’s family P_n beyond the restriction that n is odd

Establish whether Moh’s construction of the prime ideals P_n ⊂ K[[x,y,z]], defined as ker(ρ_n) for the map ρ_n: K[[x,y,z]] → K[[t]] with ρ_n(x) = t^{nm} + t^{nm+λ}, ρ_n(y) = t^{(n+1)m}, ρ_n(z) = t^{(n+2)m} and m = (n+1)/2, can be extended to even values of n, thereby removing the restriction that n is odd.

Background

In Moh’s original setup, the parameter m = (n+1)/2 is integral only when n is odd, and the construction and properties of P_n were proved under this parity restriction. The authors explicitly cite the problem of removing this restriction as an open question.

Generalizing to even n would broaden the class of prime ideals arising from such parametrizations and clarify whether the defining phenomena persist without the oddness assumption.

References

Moh's paper is an invitation to deep in these matters as he leaves some open questions, such as to avoid the hypothesis on the field to be of characteristic zero, to avoid the restriction n is odd, or to modify the exponents of the map ρ_n, so that one can find explicitly the generators of these primes ideals P_n (see, e.g., in these directions [gp1], [gp3] and [mss]).

— Binomial determinants: some closed formulae  (2603.29431 - González et al., 31 Mar 2026) in Introduction (Section 1), first paragraph, page 1