广义多项式方程参数化系统唯一解的存在性

Abhishek Deshpande, Stefan Müller
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引用次数: 0

摘要

我们考虑的是在 $n$ 正变量中,涉及具有正参数的 $m$ 单项式的广义多项式方程参数化系统(具有实指数)的解;即,$x/in/mathbb{R}^n_>$使得${A/, (c circ x^B)=0}$ 具有系数矩阵$A/in/mathbb{R}^{l次m}$,指数矩阵$B/in/mathbb{R}^{n次m}$,参数向量$c/in/mathbb{R}^m_>$,以及分量乘积$circ$。作为我们的主要结果,我们用多项式系统的相关几何对象,即 $\textit{coefficientpolytope}$ 和 $\textit{monomial dependency subspace}$ 来描述所有参数的唯一解(模为指数流形)的存在性。我们证明了唯一存在性等价于某个矩/幂映射的双射性,并利用哈达玛的全局反演定理描述了该映射的双射性。此外,我们还提供了几何对象符号向量的充要条件,从而得到了一解的多元笛卡尔符号法则。
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Existence of a unique solution to parametrized systems of generalized polynomial equations
We consider solutions to parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $x\in\mathbb{R}^n_>$ such that ${A \, (c \circ x^B)=0}$ with coefficient matrix $A\in\mathbb{R}^{l \times m}$, exponent matrix $B\in\mathbb{R}^{n \times m}$, parameter vector $c\in\mathbb{R}^m_>$, and componentwise product $\circ$. As our main result, we characterize the existence of a unique solution (modulo an exponential manifold) for all parameters in terms of the relevant geometric objects of the polynomial system, namely the $\textit{coefficient polytope}$ and the $\textit{monomial dependency subspace}$. We show that unique existence is equivalent to the bijectivity of a certain moment/power map, and we characterize the bijectivity of this map using Hadamard's global inversion theorem. Furthermore, we provide sufficient conditions in terms of sign vectors of the geometric objects, thereby obtaining a multivariate Descartes' rule of signs for exactly one solution.
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