Let $\R$ be a real closed field, $\mathcal{P},\mathcal{Q} \subset \R[X_1,...,X_k]$ finite subsets of polynomials, with the degrees of the polynomials in $\mathcal{P}$ (resp. $\mathcal{Q}$) bounded by $d$ (resp. $d_0$). Let $V \subset \R^k$ 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 $$ \displaylines{\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),}$$ where $s = \card \; \mathcal{P}$, and $$F_{d,d_0,k,k'}(j)= \textstyle\binom{k+1}{k-k'+j+1} \;(2d_0)^{k-k'}d^j\; \max{2d_0,d}^{k'-j} +2(k-j+1) .$$
In case $2 d_0 \leq d$, the above bound can be written simply as $$ \displaylines{\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k} = (sd)^{k'} d_0^{k-k'} O(1)^k} $$ (in this form the bound was suggested by J. Matousek. Our result improves in certain cases (when $d_0 \ll d$) 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 earlier in the case $d=d_0$.
The distinction between the bound $d_0$ 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.
This paper is the last paper in a series of five papers. Building on earlier papers in this series, we prove an analogue of Kuratowski's characterisation of graph planarity for three dimensions. More precisely, a simply connected 2-dimensional simplicial complex embeds in 3-space if and only if it has no obstruction from an explicit list of obstructions. This list of obstructions is finite except for one infinite family.
Yet another doubly refined enumeration of Alternating Sign Matrices
I.R.M.A., UMR 7501, Université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）
This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently introduced by Ayyer, Behrend and Fischer. We show that the number of alternating sign trapezoids associated with a Catalan object (resp. Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of the polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a c…
Alternating sign trapezoids have recently been introduced as a generalisation of alternating sign triangles. Fischer established a threefold refined enumeration of alternating sign trapezoids and provided three statistics on column strict shifted plane partitions with the same joint distribution. In this paper, we are able to add a new pair of statistics to these results. More precisely, we consider the number of $-1$s on alternating sign trapezoids and introduce a corresponding statistic on column strict shifted plane partitions that has the same distribution. More generally, we show that the…
Halved monotone triangles are a generalisation of vertically symmetric alternating sign matrices (VSASMs). We provide a weighted enumeration of halved monotone triangles with respect to a parameter which generalises the number of $-1$s in a VSASM. Among other things, this enables us to establish a generating function for vertically symmetric alternating sign trapezoids. Our results are mainly presented in terms of constant term expressions. For the proofs, we exploit Fischer's method of operator formulae as a key tool.
We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.
On refined enumerations of totally symmetric self-complementary plane partitions II
本文解决了Mills，Robbins和Rumsey在“Self-complementary totally symmetric plane partitions”（J. Combin. Theory Ser. A 42，277-292）。换句话说，我们证明了在对合下不变的移位平面划分的数目等于在垂直翻转下不变的交替符号矩阵的数目。对于一般猜想（猜想6），我们也给出了一个行列式公式，但这个行列式仍然很难计算。本文引入了两类新的多米诺平面分拆，一类与半圈对称交替符号矩阵集具有相同的基数，另一类与垂直对称交替符号矩阵集具有相同的基数.
We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen [Probab. Theory Relat. Fields 2018]. We develop general principles under which the identical scaling limits as the rank-one case can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.
