AI 中文总结
研究由特定6次齐次多项式诱导的\(\mathbb{C}^5\)到\(\mathbb{C}^5\)的多项式映射\(G\)，证明其雅可比矩阵幂幺但映射非全局单射，构造出\(G(a)=G(b)=\vec{0}\)的点对，还对梯度场零集结构分类得到37个解析复解。
AI 中文摘要
我们研究由一个6次齐次多项式\(F\)诱导的特定多项式映射\(G:\mathbb{C}^5\to\mathbb{C}^5\)，该多项式\(F\)由四个三对角调和块组成。我们证明，虽然此映射的雅可比矩阵在每一点都是幂幺的（意味着\(\det J_G(x)\equiv1\)），但该映射本身不是全局单射的。我们构造了明确的代数稀疏对\(a\neq b\)，使得\(G(a)=G(b)=\vec{0}\)。此外，我们对相应梯度场的零集结构进行了全面分类，共发现37个精确的解析复解，分布在六个不同的几何级数中。
英文摘要
We study a specific polynomial mapping G: \mathbb{C}^5 \to \mathbb{C}^5 induced by a homogeneous polynomial F of degree 6 consisting of four tridiagonal harmonic blocks. We prove that while the Jacobian matrix of this mapping is unipotent at every point (implying det J_G(x) \equiv 1), the mapping itself is not globally injective. We construct explicit algebraic sparse pairs of distinct points a \neq b that map to the identical image G(a) = G(b) = \vec{0}. Furthermore, we perform a comprehensive classification of the zero-set structure of the corresponding gradient field, uncovering a total of 37 precise analytical complex solutions split across six distinct geometric series.
CommentsThe paper has been withdrawn due to a crucial mistake in the proof
