Jacobian problem - Open Access .click

Siberian Mathematical Journal, 2022
Much work has been devoted to the Jacobian conjecture, which asks if every polynomial map $f: \mathbb C^n \to \mathbb C^n$ with non-vanishing Jacobian determinant $J_f$ is bijective with polynomial inverse. More generally one can ask under what conditions a local diffeomorphism $F: \mathbb R^n \to \mathbb R^n$ is globally injective (for example [\
V V Starkov
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Jacobian-dependent vs Jacobian-free discretizations for nonlinear differential problems

Computational and Applied Mathematics, 2020
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Jacobian Smoothing Methods for Nonlinear Complementarity Problems

SIAM Journal on Optimization, 1999
Summary: We present a new algorithm for the solution of general (not necessarily monotone) complementarity problems. The algorithm is based on a reformulation of the complementarity problem as a nonsmooth system of equations by using the Fischer-Burmeister function. We use an idea by \textit{X. J. Chen, L. Qi}, and \textit{D. F. Sun} [Math. Comput. 67,
Christian Kanzow, Heiko Pieper
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A Note on Jacobian Problem Over $$\mathbb {Z}$$

Acta Mathematica Vietnamica, 2023
The author studies the set $F(\mathbb{Z}^n)$, where $F$ are polynomial maps with $\operatorname{det}JF=1$. He proves that the number $(F^{-1}(l)\cap\mathbb{Z}^n)$ are uniformly bounded for generic lines $l\in\mathbb{Z}^n$ and $N(F(\mathbb{Z}^n),B)\leq B^{n-1}$ as $B\rightarrow +\infty$, where $F\in\mathbb{Z}[x]^n$, \(\operatorname{det ...
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