Polynomial Functions In N Indeterminates Over Finite Commutative Rings

There are two basic questions about polynomial functions over finite commutative rings:1.The sufficient and necessary conditions for a function to be a polynomial function.2.The number of polynomial function over a finite commutative ring. This paper is on the basis of the research for the first question.With the structure of finite commutative ring,we know that a finite commutative ring can be expressed by direct sum of some finite commutative local rings.So we need only to consider the judge of polynomial function over finite commutative local rings.There have so many conclusion about the first question. In this article,it extend those conclusions to several indeterminates.