Prime ideals of Moh and the characteristic of the field

TL;DR

The paper revisits Moh's construction of the Moh primes in k[[x,y,z]] and provides a characteristic-sensitive analysis of the second prime P_3 for n=3, λ=25. By reinterpreting Moh's lower bound via σ-orders and the V_r spaces, the authors give constructive procedures to obtain minimal generating sets and reveal that μ(P_3) can vary with char(k), contradicting Sally's bound and opening questions about unbounded μ in nonzero characteristic. Furthermore, they show that these minimal generators are actually Mora standard bases under the negative degree reverse lex order, strengthening both the theoretical and computational understanding of these primes. The results combine lower-bound theory with explicit constructions across characteristics, highlighting both algebraic and algorithmic aspects of prime-ideal generators in three-variable power series rings.

Abstract

We reprove and generalize a result of Moh which gives a lower bound on the minimal number of generators of an ideal in a power series ring in three variables x,y,z over a field k. As a consequence, in each characteristic of the field k, we obtain a minimal generating set for the prime ideal P of Moh corresponding to n=3. We deduce that the minimal number of generators of P might decrease depending on the characteristic of k. This contradicts a statement of Sally and leaves as an open problem to find families of prime ideals in the power series ring in the variables x,y,z with an unbounded minimal number of generators, when k has characteristic other than zero. Finally, we show that these minimal generating sets of P are standard basis with the negative degree reverse lexicographic order.

Theorems & Definitions (38)

• Remark 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• Theorem 3.2
• proof : Proof of Theorem \ref{['moh43-1']}
• proof : Proof of Theorem \ref{['moh43-2']}
• Remark 4.1
• Lemma 4.2
• proof