Integer valued polynomials

我建议的研究是在代数和数论的一般领域，并研究由值的完整性条件确定的多项式的代数，例如，具有有理系数的多项式的代数，当计算为整数时，取整数值。一个多世纪以来，这一直是代数中的一个有趣的话题，现在证明它在数学的其他领域也有应用，例如代数拓扑学和非阿基米德分析。我的研究集中在这类代数的计算方面，构建了确定基和生成集合的算法，以及相关不变量的计算。这些经常需要数论论证，并产生有趣的数论结果。这项研究的当前目标是更好地理解这种系数环不可交换时的多项式代数(例如除法代数或矩阵环)，以及这类多元多项式代数的一般计算方法。这两个方向中的任何一个方向的结果都将在代数之外有有用的应用。自20世纪70年代S以来，人们就知道这种代数与K-理论及其变种等广义上同调理论中的运算的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归结为关于多元整值多项式的问题。另外，在非阿基米德分析中，容量的计算可以表示为与此类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.