Research On Polynomial Functions And Permutation Polynomials

In this paper, we make investigations on polynomial functions and permutationpolynomials over finite commutative rings and several results are obtained.Firstly, we discuss singular permutation polynomials in several variables over thering Z/p~lZ and two results are obtained: one is a sufficient and necessary conditionfor a singular polynomial in n variables to be a permutation polynomial over Z/p~lZ isobtained; and the other is a class of classic singular permutation polynomial in severalvariables are shown. These two results generalize the relative ones of Q. Zhang, etc.,to general cases.Secondly, some discussions are given to the two basic problems on polynomialfunctions over finite commutative ring R ―when does a function over R become apolynomial one? what is the number of polynomial functions over R? We give acomplete answer to the former and our result is a generalization of the relative ones ofKempner and Q. Zhang, etc. As for the latter, we obtain a relation between the numberof polynomial functions and that of permutation polynomials over R. Additionally, wegeneralize some results to the case that functions are in several variables.Finally, we generalize the p-adic expansion of an integer to the J-adic expansionof a polynomial in the ring Z[x]. This expansion not only has theoretic sense, butalso is applied to such areas as obtaining nontrivial annihilating polynomials of thering Z/p~nZ (This plays an important role in K-Theory), and computing the number ofpolynomial functions over the ring Z/p~nZ, etc.