发表机构
Université Savoie Mont Blanc（萨瓦蒙布朗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Gale对偶与Khovanskii-Rolle论证，将任意少项式系统的正解计数转化为迭代雅可比零点与曲线分支的计数，并借助笛卡尔符号法则递归，得到仅依赖有向拟阵的上界。
AI 中文摘要
我们考虑具有 $n+k+1$ 个单项式的 $n$ 个变量中的 $n$ 个实多项式方程组。根据 Bihan 和 Sottile 的 Gale 对偶性，它们的正解对应于多面体 $\Delta\subset\mathbb{R}^k$ 中 $k$ 个方程 $\prod_ip_i^{B_{ij}}=1$ 的方程组的解，其中 $p_i$ 是仿射函数，并且 Khovanskii--Rolle 论证将其数量限制为 $\Delta$ 中迭代雅可比矩阵 $\Gamma_k,\dots,\Gamma_1$ 的公共零点数量加上某些曲线的非紧分支数量。我们将第一项限制为 Bézout 数减去排列 $\{p_i=0\}$ 的其他室中的零点数量，我们通过由面计数公式给出的边界度数从下方限制这些零点数量。曲线的分支终止于 $\Delta$ 的面上的雅可比矩阵的零点，我们通过显式简化系统在排列的平坦子空间上计数这些零点。对所有平坦子空间和室的递归从直线上的电路笛卡尔符号法则开始。我们获得正解数量的上界，该上界仅取决于系数矩阵的有向拟阵以及指数矩阵及其提升的有向拟阵。
英文摘要
We consider systems of $n$ real polynomial equations in $n$ variables with $n+k+1$ monomials. By the Gale duality of Bihan and Sottile, their positive solutions correspond to the solutions of a system of $k$ equations $\prod_ip_i^{B_{ij}}=1$ in a polyhedron $Δ\subset\mathbb{R}^k$, where the $p_i$ are affine functions, and a Khovanskii--Rolle argument bounds their number by the number of common zeros in $Δ$ of iterated Jacobians $Γ_k,\dots,Γ_1$ plus the number of noncompact branches of certain curves. We bound the first term by the Bézout number minus the numbers of zeros in the other chambers of the arrangement $\{p_i=0\}$, which we bound from below by boundary degrees given by a facet-count formula. The branches of the curves end at zeros of the Jacobians on faces of $Δ$, which we count on the flats of the arrangement through explicit reduced systems. The resulting recursion over all flats and chambers starts on lines with Descartes' rule of signs for circuits. We obtain upper bounds for the number of positive solutions which only depend on the oriented matroid of the coefficient matrix and on the oriented matroids of the exponent matrix and of its liftings.
Comments43 pages, 1 figure
