Modify the exponents in ρ_n to enable explicit generators of P_n

Develop modifications of the exponent choices in the parametrization ρ_n: K[[x,y,z]] → K[[t]] defined by ρ_n(x) = t^{nm} + t^{nm+λ}, ρ_n(y) = t^{(n+1)m}, and ρ_n(z) = t^{(n+2)m} (with m = (n+1)/2 and λ > n(n+1)m, gcd(λ,m) = 1) so that the generators of the prime ideals P_n = ker(ρ_n) can be computed explicitly.

Background

While Moh’s construction proves the existence of prime ideals P_n with notable generator bounds, obtaining explicit generating sets remains challenging for the given exponents. The authors point to Moh’s explicit identification of this issue as an open problem, suggesting that suitably altering the exponent pattern may render generators explicitly computable.

Achieving explicit generators would enhance understanding of the algebraic structure of P_n and facilitate further developments connected to binomial matrices and associated linear maps.

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