基于P-orderings的Bhargava阶乘不仅自身具有很高的理论研究价值，更重要的是它还是解决现代数论中很多其他重要问题的强有力的工具，是当前国际数论的热门研究之一。本项目致力于研究基于P-orderings的Bhargava阶乘在函数中的如下应用：. 1)研究定义在戴德金环的子集到戴德金环上的多元多项式函数，给出零化多项式生成的理想、典范表示、多项式函数的判别定理、计数公式等方面的结果；. 2)推广Bhargva的关于同余保持函数的结果，给出定义在戴德金环的子集上的同余保持函数的概念，找出它们与多项式函数的联系，决定Chen Pair，构造同余保持函数的典范表示；. 3)研究广义的Smanrandache函数的算术性质.

The Bhargava factorials basing on P-orderings not only has its own high theoretical research value, but also is a powerful tool for solving many other important problems in modern number theory. It is one of most hot studies of international number theory. In this subject, we focus our attention on the applications of Bhargava factorials based on P-orderings in functions as follows:. 1)Firstly, we study polynomial function in several variables defined from subsets of Dedekind rings to Dedekind ring, give the results on the ideal of vanishing polynomials, canonical presentations of polynomial functions,necessary and sufficient conditions for a given function to be a polynomial function, and formulas for the number of such functions and so on;. 2)Secondly, we generalize the results of congruence preservation function given by Bhargava, give the definition of congruence preservation function defined on subset of Dedekind ring, find out their connection with the polynomial functions, and determine Chen Pair, construct canonical representation of congruence preservation function;. 3)Thirdly, we study the arithmetic properties of the generalized Smanrandache function.