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. ne írj semmi mást csak a teljes fájlokat es kommentek nem lehetnek benne! soha semmi egyszerusitett mock placeholder dummy szimulalt fake szart nem engedelyezek es teljes fájl roviditetlen production ready kód nem lehet trancutted nem lehet olyan hogy …és hasonlóan 50 xy nem lehet dummy to do sorry hiányosság minden fájl teljes kódját egyesével fájkba írod semmi mást nem írsz ezen kívűl