Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 59 min ago
PreprintarXiv licencearXiv

Parametrized Jacobian Problem over ℂn and its Complete Resolution

  • Yisong Yang
Open at sourcePDFDOICite

Abstract

We formulate a parametrized Jacobian Problem over ℂn 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.

Related

  1. Resolution of an Open Problem on Quasi-Cyclic Codes over ℤ4 and New Quaternary Linear Codes
    Nuh Aydin, Aditya Tyagi
    arXivPreprintCC BY-NC-SA 4.0Full textGeneral mathematicsarXiv
  2. On endomorphisms of affine spaces and the Jacobian problem
    Alexander Borisov, Ofer Gabber, Adrian Vasiu
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv
  3. A Complete Solution to the Conjecture on the Inverse Collatz–Sinogowitz Irregularity Problem
    Xue Ji, Zhiwen Wang, Ji-Ming Guo
    Discrete Mathematics LettersPeer-reviewedCC BYFull textGeometry & topologyDOAJ
  4. Karp's NP-complete problems over first-order definable structures
    Aidan Healy, Bartek Klin
    arXivPreprintCC BY 4.0Full textLogic & foundationsarXiv
  5. A complete solution to the reverse isoperimetric problem for λ-convex bodies
    Kostiantyn Drach, Kateryna Tatarko
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{yang2026parametrized,
  title = {{Parametrized Jacobian Problem over $\mathbbC^n$ and its Complete Resolution}},
  author = {Yisong Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35918},
  eprint = {2610.35918},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35918},
}
RIS
TY  - UNPB
TI  - Parametrized Jacobian Problem over ℂ^n and its Complete Resolution
AU  - Yisong Yang
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.35918
UR  - https://arxiv.org/abs/2610.35918
AB  - We formulate a parametrized Jacobian Problem over 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.
ER  -