I need an exact mathematical formulation and a general, auto...

Prompt

I need an exact mathematical formulation and a general, automated algorithmic framework to solve the following problem in affine algebraic geometry and symbolic computation: ### Context & Background In the study of polynomial endomorphisms $F: \mathbb{K}^3 \to \mathbb{K}^3$ (where $\mathbb{K}$ is an algebraically closed field of characteristic 0, such as $\mathbb{C}$), a central open challenge related to the Jacobian Conjecture is whether there exists a regular polynomial map whose Jacobian determinant is an identically non-zero constant ($\det J(F) \in \mathbb{K}^\times$), but which fails to be injective (i.e., has non-trivial, multi-sheeted fibers). A promising geometric ansatz is based on the **tangent developable surface of an algebraic plane curve** $\boldsymbol{\gamma}(u) \subset \mathbb{K}^2$: 1. A 2D developable map parameterizing tangent lines to a curve, $(u, v) \mapsto \boldsymbol{\gamma}(u) + v \boldsymbol{\tau}(u)$ (where $\boldsymbol{\tau}(u)$ is collinear to $\boldsymbol{\gamma}'(u)$), naturally produces multi-sheeted fibers because multiple tangent lines can intersect at points outside the curve. 2. However, this classical developable map has an intrinsic singular caustic along the curve itself ($v = 0$), where its Jacobian determinant vanishes proportionally to $v$ ($\det J_{\text{dev}} \propto v$). 3. To eliminate this vanishing locus, one can apply a birational coordinate change (e.g., $v = 1/x$) that pushes the caustic $v = 0$ to infinity ($x \to \infty$). Under this substitution, the transformation Jacobian produces a pole ($\det J_{\text{trans}} \propto -1/v$), allowing the chain rule product $\det J_{\text{dev}} \cdot \det J_{\text{trans}}$ to evaluate to a non-zero constant. 4. The critical obstacle: Setting $v = 1/x$ and $u = y + 1/x$ introduces Laurent poles (singularities of the form $x^{-k}$) into the components of the map. To recover a regular polynomial map in affine space $\mathbb{K}[x, y, z]$, one must introduce a transverse shearing coordinate $X(x, y, z)$ and a polynomial vector field $\mathbf{h}(u)$ to cancel all Laurent poles via algebraic shearing: $$\mathbf{F}(u, v, X) = \begin{pmatrix} \boldsymbol{\gamma}(u) + v \boldsymbol{\tau}(u) - X \cdot \mathbf{h}(u) \\ X \end{pmatrix}$$ ### The Problem to Solve Existing attempts rely on post-hoc, hardcoded trial-and-error for specific curves (e.g., a standard parabola) and ad-hoc polynomial factorizations that do not generalize. I need a **constructive, algorithmic, and generalizable framework** that resolves the following core technical challenges: 1. **Admissible Curve Criteria:** What are the precise geometric and algebraic conditions on a plane curve $\boldsymbol{\gamma}(u) = (\gamma_1(u), \gamma_2(u))$ (its degree, singularity profile, and parameterization) such that its tangent developable can be regularized into an affine polynomial endomorphism of $\mathbb{K}^3$? 2. **Systematic Laurent Pole Cancellation:** Given an arbitrary admissible curve $\boldsymbol{\gamma}(u)$, how does one systematically construct: - The transverse shearing polynomial vector field $\mathbf{h}(u) = (h_1(u), h_2(u))^T$, and - The affine coordinate $X(x, y, z) = P(x, y) + z \cdot Q(x, y)$, such that every negative power of $x$ in $\mathbf{F}$ cancels out identically, ensuring $F_1, F_2, F_3 \in \mathbb{K}[x, y, z]$? 3. **Preservation of the Constant Jacobian Determinant:** How can the cancellation of Laurent poles be formally guaranteed without introducing higher-order variable-dependent terms into $\det J(F)$, ensuring $\det J(F) \equiv c \in \mathbb{K}^\times$ globally? 4. **Algorithmic Pipeline:** Provide a step-by-step procedural algorithm (suitable for implementation in computer algebra systems like SageMath, SymPy, or Singular) that takes any parameterized curve $\boldsymbol{\gamma}(u)$ as input and deterministically outputs the complete polynomial endomorphism $F(x, y, z)$ satisfying all constraints, along with an explicit certification of its non-injective fibers.