Integer valued polynomials

整数值多项式

• 批准号: RGPIN-2016-05308
• 负责人: Johnson, Keith
• 金额: $ 1.09万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=654736
• 关键词: Integer; valued; polynomials

My proposed research is in the general areas of algebra and number theory and studies algebras of polynomials determined by integrality conditions on their values, for example the algebra of polynomials with rational coefficients which take integer values when evaluated at integers. This has been a topic of interest within algebra for over a century which is now proving to have applications in other areas of mathematics such as algebraic topology and non-archimedean analysis. My research concentrates on computational aspects of such algebras, constructing algorithms to determine bases and generating sets and for the evaluation of associated invariants. These frequently require number theoretic arguments and produce interesting number theoretic results. The current goals of this research program are a better understanding of such algebras of polynomials when the underlying ring of coefficients is not commutative (for example division algebras or rings of matrices) and general computational methods for such algebras of polynomials in several variables. Results in either of these directions would have useful applications outside of algebra. It has been known since the 1970's that there is a close connection between such algebras and Hopf algebras of operations in generalized cohomology theories such as K-theory and its variants and that homological calculations for these theories can be cast as problems about integer valued polynomials in several variables. Also in non-archimedean analysis the computation of the capacity can be expressed as the evaluation of a limit of certain invariants of associated to such algebras.

我建议的研究是在一般领域的代数和数论和研究代数的多项式所确定的完整性条件对他们的价值观，例如代数的多项式与合理的系数采取整数值时，评估在整数。这一直是一个话题的兴趣在代数超过一个世纪，现在证明有应用在其他领域的数学，如代数拓扑和非阿基米德分析。我的研究集中在计算方面的代数，构建算法，以确定基地和发电机组和评估相关的不变量。这些经常需要数论参数和产生有趣的数论结果。这个研究计划的当前目标是更好地理解这样的多项式代数时，系数的基础环是不交换的（例如除法代数或矩阵环）和一般的计算方法，这样的代数多项式在几个变量。这两个方向的结果在代数之外都有有用的应用。自20世纪70年代以来，人们已经知道，在广义上同调理论（如K-理论及其变体）中，这种代数与Hopf代数之间存在密切联系，并且这些理论的同调计算可以被视为关于多变量整数值多项式的问题。在非阿基米德分析中，容量的计算也可以表示为对与这种代数相关的某些不变量的极限的评估。