Let k be an algebraically closed field of characteristic zero and B a factorial affine k-domain equipped with a locally nilpotent derivation δ. We investigate B when there exists an element z ∈ B such that δ(z) = α^p for p ≥ 1 and a prime element α of A = Ker δ. One such example of particular interest is the coordinate ring of an affine pseudo-n-space, which is defined as a smooth affine variety X equipped with a faithfully at morphism q : X → 𝔸¹ such that q^–1(𝔸\( {}_{\ast}^1 \)) ≅ 𝔸\( {}_{\ast}^1 \) × 𝔸 ^n–1 and q^*(0) is irreducible and reduced where 𝔸\( {}_{\ast}^1 \) = 𝔸¹ \ {0}. In [9], a criterion was given for an affine pseudo-3-space to be isomorphic to a hypersurface of form x^py – g(x; t; z) = 0 in 𝔸⁴. Such a hypersurface is called a hypersurface of Danielewski type and studied in [17], [16]. In this article, under the condition that A/αA is factorial, we describe B in terms of equivariant affine modification developed by Kaliman and Zaidenberg [15] and give a criterion for B to be isomorphic to the residue ring A[Y;Z]/(α^pY – g(Z)) for p ≥ 1 and an irreducible polynomial g(Z) ∈ A[Z]\A. As a consequence, we obtain a criterion for an affine pseudo-n-space to be isomorphic to a hypersurface of Danielewski type for n ≥ 3.

令k为特征零的代数闭域，B为阶乘仿射k域，具有局部幂零导数δ。我们研究B当存在一个元素z ∈ B使得δ (z) = α ^p对于 p ≥ 1 和A的素元素 α = Ker δ。一个特别有趣的例子是仿射伪n空间的坐标环，它被定义为一个光滑的仿射簇X，具有忠实地在态射q : X → 𝔸¹使得q ^–1 (𝔸 \( {}_{\ast}^1 \) ) ≅ 𝔸 \( {}_{\ast}^1 \) × 𝔸 ^{n –1}和q ^* (0) 是不可约的并减少了 𝔸 \( {}_{\ast}^1 \) = 𝔸 ¹ \ {0}。在 [9] 中，给出了仿射伪 3 空间与形式x ^p y – g ( x ; t ; z ) = 0 in 𝔸 ⁴的超曲面同构的标准。这种超曲面称为 Danielewski 类型的超曲面，并在 [17]、[16] 中进行了研究。在本文中，在A/α A是阶乘，我们用 Kaliman 和 Zaidenberg [15] 开发的等变仿射修饰来描述B，并给出B与残基环A [ Y同构的标准；Z ]/(α ^p Y – g ( Z )) 对于p ≥ 1 和不可约多项式g ( Z ) ∈ A [ Z ]\ A。因此，我们获得了一个仿射伪n空间与n ≥ 3的 Danielewski 类型超曲面同构的标准。