论文标题
无限的临界点方向超平面
Critical Points at Infinity for Hyperplanes of Directions
论文作者
论文摘要
几个变量（ACSV）中的分析组合学分析了$ \ mathbf {r} $ in方向上的meromormormormorphic生成函数的系数的渐近生长$ f = g/h $。它在极点品种上使用莫尔斯理论$ v：= \ {h = 0 \} \ subseteq（\ mathbb {c}^*）$ f $ of $ f $ of $ f $的$ f $，以通过\ textit {height fluct {height flog} flow \ textit {height} = h = h = h = h = h = h = h = h = h = h = - \ sum_ {j = 1}^d r_j \ log | z_j | $，将$ t $的同源性分解归为\ textit {critical {critical {critical {critical {critical {crigital Points}的$ h $ on $ v $。当高度函数不是合适的地图时，变形可以在有限高度下流到无穷大。 This happens only in the presence of a critical point at infinity (CPAI): a sequence of points on $V$ approaching a point at infinity, and such that log-normals to $V$ converge projectively to $\mathbf{r}$.如果高度函数也收敛到有限值，则CPAI称为\ textIt {高高}。本文研究了所有CPAI是否均已升高，以及CPAI的方向。我们通过检查序列的序列与多项式$ h $的牛顿polytope $ \ Mathcal {p} $定义的曲面紧凑型的面孔进行研究。在一般满意的条件下，序列的日志正态符号的任何投影限制都融合到face $ f $的情况下，必须平行于$ f $；这意味着CPAI必须始终升高，并且只能在与$ \ Mathcal {p} $的某些面上平行的方向上发生。当这种通用条件失败时，我们在平稳性的情况下显示，Codimension-1 face $ f $中的一个点仍然只能是平行于$ f $的方向的CPAI，并且Codimension-2 Face的方向可以是较大的集合，可以显然计算出明确的计算，并且仍然具有正编辑。
Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of the coefficients of a meromorphic generating function $F = G/H$ in a direction $\mathbf{r}$. It uses Morse theory on the pole variety $V := \{ H = 0 \} \subseteq (\mathbb{C}^*)^d$ of $F$ to deform the torus $T$ in the multivariate Cauchy Integral Formula via the downward gradient flow for the \textit{height} function $h = h_{\mathbf{r}} = -\sum_{j=1}^d r_j \log |z_j|$, giving a homology decomposition of $T$ into cycles around \textit{critical points} of $h$ on $V$. The deformation can flow to infinity at finite height when the height function is not a proper map. This happens only in the presence of a critical point at infinity (CPAI): a sequence of points on $V$ approaching a point at infinity, and such that log-normals to $V$ converge projectively to $\mathbf{r}$. The CPAI is called \textit{heighted} if the height function also converges to a finite value. This paper studies whether all CPAI are heighted, and in which directions CPAI can occur. We study these questions by examining sequences converging to faces of a toric compactification defined by a multiple of the Newton polytope $\mathcal{P}$ of the polynomial $H$. Under generically satisfied conditions, any projective limit of log-normals of a sequence converging to a face $F$ must be parallel to $F$; this implies that CPAI must always be heighted and can only occur in directions parallel to some face of $\mathcal{P}$. When this generic condition fails, we show under a smoothness condition, that a point in a codimension-1 face $F$ can still only be a CPAI for directions parallel to $F$, and that the directions for a codimension-2 face can be a larger set, which can be computed explicitly and still has positive codimension.
