论文标题
真实多种疗法曲面和同源稳定性的拓扑结构
Topology of real multi-affine hypersurfaces and a homological stability property
论文作者
论文摘要
令$ \ mathrm {r} $为一个真正的封闭字段。我们证明，由$ \ mathrm {r}^n $在由$ 2^{d-1} $界限的多项式多项式定义的$ \ mathrm {r}^n $中的真实超曲面的半代数连接组件的数量。该界限是锋利的，并且独立于$ n $（而不是$ d（2d -1）^{n -1} $的经典限制，这是由$ \ mathrm {r}^n $的任意多项式$ d $ in TEM $ d $ n $ d $ n $ d $ d $ n $ {r}^n $所定义的，这是因为Petrovski {r}^n $ wy th y Mile {此外，我们显示存在$ c> 1 $，因此给定一个序列$（b_n）_ {n> 0} $，其中$ b_n $是$ \ mathrm {rmathrm {r}^n $积极辐射的封闭球，存在hyperSurfaces $（v_n）_ {n_> 0} $ aftersurfaces $ aftermement propern $ aft in n _> 0} $ 4 $ \ sum_ {i \ leq 5} b_i（v_n \ cap b_n）> c^n $，其中$ b_i（\ cdot）$表示带有有理系数的$ i $ th betti编号。最后，作为本文的主要结果的应用，我们验证了由Basu和Riener引起的代表性稳定性，这是对对称真实代数集的共同体学模块，用于一系列新的和更大的对称真实代数集，该集胜过以前的知名度。
Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the classical bound of $d(2d -1)^{n-1}$ on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree $d$ in $\mathrm{R}^n$ due to Petrovski{\uı} and Ole{\uı}nik, Thom and Milnor). Moreover, we show there exists $c > 1$, such that given a sequence $(B_n)_{n >0}$ where $B_n$ is a closed ball in $\mathrm{R}^n$ of positive radious, there exist hypersurfaces $(V_n)_{n_>0}$ defined by symmetric multi-affine polynomials of degree $4$, such that $\sum_{i \leq 5} b_i(V_n \cap B_n) > c^n$, where $b_i(\cdot)$ denotes the $i$-th Betti number with rational coeffcients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.
