非例外单项情形下Villamayor算法的复杂性
Complexity of Villamayor's algorithm in the non exceptional monomial case
摘要：
我们研究单项式理想，总是当地的单项式，像一个合理的第一步，估计一般的数量monoidal变换的Villamayor的算法的决议的奇异性。一个单项理想$<X_1^{a_1}\cdot ... \cdot X_n^{a_n}>$的决议是有趣的，由于其识别与特定的复曲面问题$<Z^c- X_1^{a_1}\cdot ... \cdot X_n^{a_n}>$。在特殊情况下，当所有的指数$a_i$都大于或等于临界值$c$，我们构造的最大分支的决议树提供了一个上限涉及Catalan数的部分和。这种情况称为“最小余维情况”。卡塔兰数的部分和（从$1,2,5,...$开始）是$1,3,8,22,...$这些部分和在组合学中是众所周知的，并且计算所有具有$n+1$边的有序树中从根开始的路径的数量。卡塔兰数出现在许多组合问题中，计算在$n+1$字母的单词中插入$n$括号对的方法的数量，具有$n+1$顶点的平面树，$... $等。非最小情况下，当存在一些指数$a_{i_0}$小于$c$时，将被称为“更高余维的情况”。在这种情况下，仍然没有解决，我们举一个例子来说明最重要的麻烦。计算的例子已经在这两种情况下，研究的分辨率不变量的行为有所帮助。计算已在奇异（见\cite{sing}）使用\emph{desing}包由G。Bodnár and J. Schicho，see \cite{lib} .
英文摘要：
We study monomial ideals, always locally given by a monomial, like a reasonable first step to estimate in general the number of monoidal transformations of Villamayor's algorithm of resolution of singularities. The resolution of a monomial ideal $<X_1^{a_1}\cdot ... \cdot X_n^{a_n}>$ is interesting due to its identification with the particular toric problem $<Z^c- X_1^{a_1}\cdot ... \cdot X_n^{a_n}>$. In the special case, when all the exponents $a_i$ are greater than or equal to the critical value $c$, we construct the largest branch of the resolution tree which provides an upper bound involving partial sums of Catalan numbers. This case will be called ``minimal codimensional case''. Partial sums of Catalan numbers (starting $1,2,5,...$) are $1,3,8,22,...$ These partial sums are well known in Combinatorics and count the number of paths starting from the root in all ordered trees with $n+1$ edges. Catalan numbers appear in many combinatorial problems, counting the number of ways to insert $n$ pairs of parenthesis in a word of $n+1$ letters, plane trees with $n+1$ vertices, $... $, etc. The non minimal case, when there exists some exponent $a_{i_0}$ smaller than $c$, will be called ``case of higher codimension''. In this case, still unresolved, we give an example to state the foremost troubles. Computation of examples has been helpful in both cases to study the behaviour of the resolution invariant. Computations have been made in Singular (see \cite{sing}) using the \emph{desing} package by G. Bodnár and J. Schicho, see \cite{lib}.
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