摘要翻译：
我们研究总是由单项式局部给出的单项式理想，就像估计Villamayor奇异性分解算法的单项式变换个数的合理的第一步。一个单项式理想$<x_1^{a_1}\cdot...\cdot x_n^{a_n}>$的解是有趣的，因为它与特殊的toric问题$<z^c-x_1^{a_1}\cdot...\cdot x_n^{a_n}>$有关。在特例中，当所有指数$a_i$都大于或等于临界值$c$时，我们构造了分辨率树的最大分支，该分支提供了一个包含部分Catalan数和的上界。这种情况称为“最小余维情况”。加泰罗尼亚数的部分和（开始是$1,2,5,…$)是$1,3,8,22,…$这些部分和在组合学中是众所周知的，它计算所有有$n+1$边的有序树中从根开始的路径数。加泰罗尼亚数出现在许多组合问题中，计算在一个由$N+1$字母组成的单词中插入$N$对括号的方法的数量，具有$N+1$顶点的梧桐树，$······当存在某个指数$a_{i_0}$小于$c$时，非极小情形称为“高余维情形”。在这种情况下，仍然没有解决，我们举一个例子来陈述最重要的问题。实例计算在这两种情况下都有助于研究分辨率不变量的行为。G.Bodn\'ar和J.Schicho使用\emph{desing}包进行了奇异（参见\cite{sing}）计算，参见\cite{lib}。
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英文标题：
《Complexity of Villamayor's algorithm in the non exceptional monomial
  case》
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作者：
Rocio Blanco
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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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. Bodn\'ar and J. Schicho, see \cite{lib}.
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PDF链接：
https://arxiv.org/pdf/0704.3416