利用Newton多胞形和圈数研究全局非纯多项式映射
On Globally Diffeomorphic Polynomial Maps via Newton Polytopes and Circuit Numbers
摘要：
本文通过研究多项式平方和$\|F\|_2^2$在无穷远处的Newton多面体的性质，分析了多项式映射$F:\mathbb{R}^n\rightarrow\mathbb{R}^n$的整体同形性.这使我们能够确定一类多项式映射$F$，其在$\mathbb{R}^n$上的全局同形性等价于其雅可比行列式$\text{det }JF$在$\mathbb{R}^n$上无处消失。换句话说，我们确定了一类多项式映射的真正的雅可比猜想，这被证明是假的一般，仍然成立。
英文摘要：
In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $\|F\|_2^2$. This allows us to identify a class of polynomial maps $F$ for which their global diffeomorphism property on $\mathbb{R}^n$ is equivalent to their Jacobian determinant $\text{det }JF$ vanishing nowhere on $\mathbb{R}^n$. In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.
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