极大加代数上形式多项式的带余除法

THE DIVISION ALGORITHM FOR FORMAL POLYNOMIALS IN MAX-PLUS ALGEBRA

研究极大加代数上形式多项式的带余除法.引入形式多项式可除的概念,给出可除的一些性质.在此基础上,研究二次凹多项式与次数小于2的多项式之间的可除关系,给出两个多项式可除的一个充分必要条件,商式和余式唯一的一个充分必要条件以及商式和余式的求法.举例说明凹多项式之间的可除关系与多项式函数之间的可除关系的等价性.利用这个带余除法可计算极大加代数上循环码的循环移位.

The division algorithm of formal polynomials in max-plus algebra is investigated in this paper.We introduce the concept of divisible for formal polynomials and give some of its properties.On this basis,we consider the divisibility relationship between any quadratic concavified polynomial and any formal polynomials whose degree is less than 2.The necessary and sufficient condition of the quadratic concavified polynomial to be divisible by another formal polynomial is presented.We also give the necessary and sufficient condition of the quotient and remainder to be unique.In addition,a method to calculate the quotient and remainder which satisfies the division algorithm is introduced.Two numerical examples are used to illustrate that the divisibility of formal polynomials is equivalent to the divisibility of polynomial functions in max-plus algebra.Using this division algorithm,one may calculate the circular shift of the cycle code over max-plus algebra.