Word maps and Waring type problems
Word maps and Waring type problems
复制标题
字词地图和 Waring 类型问题
DOI：
10.1090/s0894-0347-08-00615-2
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发表时间：
2007
影响因子：
3.9
通讯作者：
A. Shalev
中科院分区：
文献类型：
--
作者：
M. Larsen;A. Shalev
华林问题的问题是，是否每个自然数都是g（k）的fc次幂之和（其中g是一个合适的函数）。1909年，休伯特肯定地解决了这个问题。优化g（k）一直是加性数论的中心问题（更多细节和背景见[Na]）。最近有相当大的兴趣在群论类似的这种现象，其目的是目前的群元素作为短产品的某些“特殊”的元素。这些特殊元素可以是群中一般字w的幂或乘子或值，也可以是群中特殊共轭类的元素。这样做的动机有时是拓扑的。设G是一个（拓扑）生成的pro-p群.利用交换子子群G1的元素的表达式作为交换子的有界积，Serre证明了G'是闭的，并推导出G的每个有限指数子群是开的。Nikolov和Segal [NSI，NS 2]最近的一个深入结果表明，这对每个n-生成的profinite群都成立。再次证明的核心是提出集团元素作为短产品的价值观的某些话。
Waring's problem asks whether every natural number is a sum of g(k) fcth powers (where g is a suitable function). This was solved affirmatively by Hubert in 1909. Optimizing g(k) has been a central problem in additive number theory ever since (see [Na] for more detail and background). Recently there has been considerable interest in group theoretic analogues of this phenomenon, where the aim is to present group elements as short products of certain "special" elements. These special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group. The motivation for this is sometimes topological. Let G be a (topologically) finitely generated pro-p group. Using expressions of elements of the commutator subgroup G1 as bounded products of commutators, Serre showed that G' is closed and deduced that every finite index subgroup of G is open. A recent deep result of Nikolov and Segal [NSI, NS2] shows that this holds for every finitely generated profinite group. Again the core of the proof is presenting group elements as short products of values of certain words.