论文标题
几个典型的超曲面的连接组件数量的新界限
New bounds for the number of connected components of fewnomial hypersurfaces
论文作者
论文摘要
我们证明，由$ \ mathbb {r}^n $的积极矫正处的相关组件数量由由$ d + k + 1 $单元定义的$ \ mathbb {r}^n $数量改善先前已知的界限。我们通过表明由$ n $变量的真实多项式和$ d + 3 $单元所定义的平滑性超表面来完善这种限制，以$ n $变量的$ \ lfloor（d-1）/2 \ rfloor + 3 $ connected组件的正质量$ \ \ \ \ \ mathbbbb，rfloor + 3 $连接的组件。我们在$ 2 $变量中呈现一个明确的多项式，并带有$ 5 $单元的单元，该变量定义了曲线，其中有三个连接的组件在正矫正中，这表明我们的界限是$ d = 2 $（和任何$ n $）的锐利。我们的结果适用于具有实际指数向量的多项式。
We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lfloor(d-1)/2\rfloor + 3$ connected components in the positive orthant of $\mathbb{R}^n$. We present an explicit polynomial in $2$ variables with $5$ monomials which defines a curve with three connected components in the positive orthant, showing that our bound is sharp for $d = 2$ (and any $n$). Our results hold for polynomials with real exponent vectors.
