Topic
Nested radical
About: Nested radical is a research topic. Over the lifetime, 12 publications have been published within this topic receiving 97 citations.
Papers
TL;DR: An algorithm for determining a linearly independent basis for a set of radicals (nested or not), which allows elementary techniques to be used for arithmetic operations on expressions involving elements of this set, and a structure theorem that provides a sufficient condition for a nested radical to be expressed in terms of radicals of lower nesting level.
39 citations
TL;DR: In this article, a primitive lth root of unity is respresented by its symbol $\zeta_l $, rather than as a nested radical, and the algorithms require computing the splitting field of the minimal polynomial of
Abstract: Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then necessary and sufficient conditions for a denesting are given, and the algorithm computes a denesting of $\alpha $ when it exists. If the base field does not contain all roots of unity, then it is shown how to compute a denesting that is within one of optimal over the base field adjoining a single root of unity. Throughout this paper, a primitive lth root of unity is respresented by its symbol $\zeta_l $, rather than as a nested radical. The algorithms require computing the splitting field of the minimal polynomial of $\alpha $ over k, and have exponential running time.
37 citations
24 Oct 1992
TL;DR: The author presents a simple condition when nested radical expressions of depth two can be denested using real radicals or radicals of some bounded degree and describes the structure of these denestings and determines an upper bound on the maximum size of a denesting.
Abstract: The author presents a simple condition when nested radical expressions of depth two can be denested using real radicals or radicals of some bounded degree. He describes the structure of these denestings and determines an upper bound on the maximum size of a denesting. Also for depth two radicals he describes an algorithm that will find such a denesting whenever one exists. Unlike all previous denesting algorithms the algorithm does not use Galois theory. In particular, he avoids the construction of the minimal polynomial and splitting field of a nested radical expression. Thus he can obtain the first denesting algorithm whose run time is at most, and in general much less, than polynomial in description size of the minimal polynomial. The algorithm can be used to determine non-trivial denestings for expressions of depth larger than two. >
11 citations
TL;DR: This work fills a lacuna in Zippel's proof, and shows that his sufficient condition is also necessary for a nested radical to be expressed in terms of radicals of lower nesting depth.
10 citations
15 Sep 1997
TL;DR: This work shows how to compute an optimal or near optimal depth denesting of α by a nested radical that involves only D th roots, where D is an arbitrary multiple of d .
Abstract: Given a nested radical α involving only d th roots, we show how to compute an optimal or near optimal depth denesting of α by a nested radical that involves only D th roots, where D is an arbitrary multiple of d . As a special case the algorithm can be used to compute denestings as in [9]. The running times of the algorithms are polynomial in the description size of the splitting field for α.
3 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2015 | 1 |
| 2014 | 1 |
| 2013 | 1 |
| 2008 | 1 |
| 2007 | 1 |
| 2000 | 1 |