可实现符号条件的半代数连通分支个数的渐近紧界

On the speed of algebraically defined graph classes

The speed of a class of graphs counts the number of graphs on the vertex set $\lbrace 1,\dots, n\rbrace$ inside the class as a function of $n$. In this paper, we investigate this function for many classes of graphs that naturally arise in discrete geometry, for example intersection graphs of segments or disks in the plane. While upper bounds follow from Warren's theorem (a variant of a theorem of Milnor and Thom), all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving an essentially tight lower bound for the nu…

Bezout不等式的一个实类比与符号条件的连通分支数

On a real analogue of Bezout inequality and the number of connected components of sign conditions

设$\mathrm{R}$是实闭域，且$Q_1, \ldots, Q_{\ell} \in \mathrm{R}[X_1, \ldots,X_k]$使得对于每个$i, 1 \leq i \leq \ell$，$\mathrm{deg} (Q_i) \leq d_i$。对于$1 \leq i \leq \ell$，用$\mathcal{Q}_i = \{Q_1, \ldots, Q_i \}$表示，$V_i$是由$\mathcal{Q}_i$定义的实簇，$k_i$是$V_i$的实维数上界（按照惯例$V_0 = \mathrm{R}^k$和$k_0 = k$）。还假设\[ 2 \leq d_1 \leq d_2 \leq \frac{1}{k + 1} d_3 \leq \frac{1}{(k + 1)^2} d_4 \leq \cdots \leq \frac{1}{(k + 1)^{\ell - 3}} d_{\ell - 1} \leq \frac{1}{(k + 1)^{\ell - 2}} d_{\ell}, \]和$\ell \leq k$。证明了$V_{\ell}$的半代数连通分支的个数由\[ O (k)^{2 k} \left(\prod_{1 \leq j < \ell} d_j^{k_{j - 1} - k_j} \right) d_{\ell}^{k_{\ell …

簇上符号条件连通分支数的精化界

Refined bounds on the number of connected components of sign conditions on a variety

设$\R$是一个实闭域，$\mathcal{P},\mathcal{Q} \subset \R[X_1,...,X_k]$是多项式的有限子集，其中多项式的次数在$\mathcal{P}$中（分别为. $\mathcal{Q}$）由$d$（分别为PDMATHAMR）。设$V \subset \R^k$是由$\mathcal{Q}$中的多项式定义的实代数簇，并假设$V$的实维度以$k'$为界。我们证明了$\mathcal{P}$族在$V$上的所有可实现符号条件的实现的半代数连通分支的个数由$$ \displaylines{\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),}$$有界，其中$s = \card \; \mathcal{P}$和$$F_{d,d_0,k,k'}(j)= \textstyle\binom{k+1}{k-k'+j+1} \;(2d_0)^{k-k'}d^j\; \max{2d_0,d}^{k'-j} +2(k-j+1) .$$在$2 d_0 \leq d$的情况下，上述界可以简单地写为$$ \displaylines{\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k} = (sd)^{k'} d_0^{k-k'} O(1)^k} $$（在这种形…

图的连通划分计数

Counting Connected Partitions of Graphs

受Győri-Lovász定理启发，本文研究了图的连通划分计数问题，分别针对边划分和点划分情形，证明了关于可实现连通子图边数或顶点数的无序正整数解个数的多个紧致下界。