Integer valued polynomials

我建议的研究是在代数和数论的一般领域，并研究由其值的完整性条件决定的多项式代数，例如，在整数计算时取整数值的有理系数多项式的代数。一个多世纪以来，这一直是代数领域的一个有趣的话题，现在证明它在数学的其他领域也有应用，如代数拓扑和非阿基米德分析。我的研究集中在这些代数的计算方面，构建算法来确定基和生成集以及相关不变量的评估。这些经常需要数论论证，并产生有趣的数论结果。本研究计划的当前目标是更好地理解当系数环不可交换时的多项式代数（例如除法代数或矩阵环）和多变量多项式代数的一般计算方法。这两个方向的结果在代数之外都有很好的应用。自20世纪70年代以来，人们已经知道这些代数与广义上同调理论（如k理论及其变体）中运算的Hopf代数之间有着密切的联系，并且这些理论的同调计算可以被看作是关于几个变量的整数值多项式的问题。同样，在非阿基米德分析中，容量的计算可以表示为与这些代数相关的某些不变量的极限的求值。

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.