AI 中文总结
该研究构造特征2下的二维多项式自同态，证明其雅可比行列式为1、扩域可分且一般次数与特征互素，给出可分雅可比猜想的二维反例，还推导了相关映射并记录形式化复现信息。
AI 中文摘要
设k是F₂的代数闭包，我们研究仿射平面的多项式自同态F=(P,Q)，其中P=x+x²y+x⁴+x⁶y²，Q=y+x⁵+x⁶y+x⁷y²+x⁸y³。其雅可比行列式为1，而三个不同的点(0,1)、(1,0)和(1,1)具有相同的像。我们证明[k(x,y):k(P,Q)]=3，且该扩域是可分的。因此，尽管F不是自同构，但一般次数与特征互素，这为特征2下的可分雅可比猜想提供了一个二维反例。证明过程用到从隐藏三次式的显式恢复、在实际目标域上的不可约性，以及函数域嵌入与几何一般纤维之间的桥梁。我们还给出一个显式图表示，证明F是平展的，并推导了来自Irit Huq-Kuruvilla的三变量映射的坐标置换形式的映射。附录记录了Lean形式化的精确范围和证据边界，以及独立的Harmonic Aristotle复现。
英文摘要
Let k be the algebraic closure of F_2. We study the polynomial endomorphism F=(P,Q) of the affine plane, where P=x+x^2 y+x^4+x^6 y^2 and Q=y+x^5+x^6 y+x^7 y^2+x^8 y^3. Its Jacobian determinant is 1, while the three distinct points (0,1), (1,0), and (1,1) have the same image. We prove that [k(x,y):k(P,Q)]=3 and that the extension is separable. Thus the generic degree is prime to the characteristic although F is not an automorphism, giving a dimension-two counterexample to the separable Jacobian conjecture in characteristic two. The proof uses explicit recovery from a hidden cubic, irreducibility over the actual target field, and a bridge between function-field embeddings and the geometric generic fiber. We also give an explicit graph presentation proving that F is etale and derive the map from a coordinate-permuted form of a three-variable map of Irit Huq-Kuruvilla. An appendix records the precise scope and evidence boundaries of a Lean formalization and an independent Harmonic Aristotle replay.
Comments10 pages. Includes a Lean 4 formalization appendix and reproducibility receipts. No external figures
