Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety
Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety
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各种符号条件连通分量数量的细化界限
DOI：
10.1007/s00454-011-9391-3
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发表时间：
2011
影响因子：
0.8
通讯作者：
S. Basu
中科院分区：
文献类型：
--
作者：
Sal Barone;S. Basu
摘要设${\textnormal {R}}$为实闭域，$\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]$为多项式的有限子集，多项式的阶数为$\mathcal{P}$(对应于， $\mathcal{Q}$)，以d为界。， 10)。设$V \subset {\textnormal {R}}^{k}$是由$\mathcal{Q}$中的多项式定义的实代数变量，并假设V的实维数以k '为界。证明了族$\mathcal{P}$在V上的所有可实现符号条件的实现的半代数连通分量的个数有
$$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$$ where $s = \operatorname {card}\mathcal{P}$, and
$$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$$
当2d0≤d时，上界可简写为
$$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$$（这种形式的边界由Matousek在2011年提出）。我们的结果在某些情况下（当0≪d）改善了已知的极限
$$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$$与Basu et al. （Proc. Am.）证明的数字相同。数学。社会学。133(4):965-974,2005)的情况下，d= 0。定义变量V的多项式度的界d0和出现在新界$\mathcal{P}$中多项式度的界d之间的区别是由离散几何中的几个应用驱动的（Guth和Katz在arXiv:1011.4105v1[数学]）。[j], 2011；Kaplan et al. [j] .中国科学d辑（英文版）；[j], 2011；数学学报（自然科学版）[j]；[j], 2011；[j] .中国科学：地球科学[j] .北京：北京大学。Co], 2011)。
AbstractLet ${\textnormal {R}}$ be a real closed field, $\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]$ finite subsets of polynomials, with the degrees of the polynomials in $\mathcal{P}$ (resp., $\mathcal{Q}$) bounded by d (resp., d0). Let $V \subset {\textnormal {R}}^{k}$ be the real algebraic variety defined by the polynomials in $\mathcal{Q}$ and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family $\mathcal{P}$ on V is bounded by
$$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$$ where $s = \operatorname {card}\mathcal{P}$, and
$$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$$
In case 2d0≤d, the above bound can be written simply as
$$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$$ (in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d0≪d) the best known bound of
$$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$$ on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d0.The distinction between the bound d0 on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in $\mathcal{P}$ that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1 [math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1 [math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2 [math.CO], 2011; Zahl in arXiv:1104.4987v3 [math.CO], 2011).