The Division For Formal Polynomials And Linear Codes For Coding In Max-Plus Algebra

Max-plus algebra is an algebraic system with important theoretical significance and application values.The polynomial is one of basic research objects in algebra. The max-plus linear system theory has developed and improved continuously, however, the polynomial theory in max-plus algebra is seldom studied. The division with reminder is an important method in number theory and polynomial theory. It plays an important role in Euclidean algorithm, factorization, solving polynomial equations, and decomposing rational functions.We will study the division and the division with reminder of polynomials in max-plus algebra. Under this basis, this paper is the first to study the linear codes of coding in max-plus algebra.First, we investigate the relationship between formal polynomials and polynomial functions, and prove that there exists an evaluation homomorphism between the formal polynomial idempotent algebra and the polynomial function idempotent algebra. Second, we study the properties of the concavified polynomials and find that any concavified polynomial has a full support. Third, we introduce the concepts of divisible, quotient and remainder, and give some of their properties. By using the characteristic of concavified polynomials having full supports and the monotonicity of differences between adjoining monomials’ coefficients, we consider the divisibility relation between any quadratic concavified polynomial and any formal polynomial with degree less than 2. The necessary and sufficient condition that the quadratic concavified polynomial is divisible by any other formal polynomial with degree 1 is presented. The necessary and sufficient condition for quotient and remainder to be unique is given. In addition, we also give a method to calculate the quotient and remainder satisfying the division with remainder.The concavified polynomials have special meanings for us to study the division of polynomial functions. By using the properties of the evaluation homomorphism, we prove that two polynomial functions are divisible if the two corresponding concavified polynomials are divisible. According to the calculation formula, we can calculate the quotient and remainder that yield from the division between any quadratic polynomial function and any polynomial function with degree 1. On the other hand, we also exemplify that two polynomial functions are indivisible if the two corresponding concavified polynomials are indivisible.Finally, we consider the linear codes for coding in max-plus algebra. We introduce the concept of max-plus linear codes and max-plus cycle codes, and give the polynomial description for linear codes in max-plus algebra, which is called the max-plus code polynomial, and present some numerical examples. The division of polynomials in max-plus algebra can be used to calculate the cyclic shift of max-plus cycle codes.