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摘要翻译：
本文致力于求单位根中的多项式方程的解。由S.Lang推测，由M.Laurent证明，所有这样的解都可以用有限个称为最大扭转陪集的参数族来描述。我们得到了复代数$n$-torus${\mathbb G}_{\rm m}^n$的代数子簇上的最大扭转陪集个数的新的显式上界。与以前给出多项式增长在定义多项式的最大总次上的界相比，我们的结果是构造性的。这使得我们得到了一个确定代数子簇${\mathbb G}_{\rm m}^n$上最大扭陪集的新算法。
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英文标题：
《Solving algebraic equations in roots of unity》
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作者：
Iskander Aliev and Chris Smyth
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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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：
This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic $n$-torus ${\mathbb G}_{\rm m}^n$. In contrast to earlier works that give the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This allows us to obtain a new algorithm for determining maximal torsion cosets on an algebraic subvariety of ${\mathbb G}_{\rm m}^n$.
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PDF链接：
https://arxiv.org/pdf/0704.1747
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