arXiv:2610.35918v1 Announce Type: new Abstract: We formulate a parametrized Jacobian Problem over $\mathbb{C}^{n}$ by incorporating independent scale parameters into the linear part of a polynomial map via polar decomposition. We prove that requiring the Jacobian determinant to remain constant uniformly across the parameter space decouples the Jacobian equation into a system enforcing the vanishing of all principal minors of the perturbation matrix. This system strictly forces lower triangularization up to variable permutations, establishing a necessary and sufficient condition for polynomial invertibility and providing an explicit, canonical scheme for constructing the inverse map in every dimension.
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