多项式型迭代方程的多项式解

摘要/Abstract

摘要：

关于多项式型迭代方程的绝大多数结果都是在已知函数为单调函数情形下给出的.该文研究了多项式型迭代方程在已知函数为特殊的非单调函数-多项式函数情形下的多项式解.首先，在一维情形下，利用计算机代数系统SINGULAR分解代数簇的方法分别给出了二次和三次多项式型迭代方程有二次多项式解的充分必要条件，以及解的具体形式，并进一步给出计算多项式解的算法.最后利用一维情形的思想研究了二维情形下多项式型迭代方程的多项式解，在已知函数为平面二次齐次多项式映射时，得到了二次多项式型迭代方程有平面二次保次多项式解的几个充要条件.

关键词: 多项式型迭代方程, 首项系数问题, 多项式解, 代数簇, 最小不可约分解

Abstract:

Most of known results for the polynomial-like iterative equation were given for monotone functions. In this paper, we discuss this equation for a polynomial function, which is non-monotonic. In one-dimensional case, we apply the method of computer algebra system SINGULAR decomposing algebraic varieties to find a sufficient and necessary condition for the polynomial-like iterative equation of orders 2 and 3 having quadratic polynomial solutions respectively and give quadratic polynomial solutions of both two equations. Then we give a procedure for computing polynomial solutions of the polynomial-like equation. In two-dimensional case, applying the idea of one-dimensional case we obtain several sufficient and necessary conditions for second order polynomial-like iterative equation having quadratic degree-preserving polynomial solutions when the given function is a two-dimensional homogeneous polynomial mapping of degree 2.

Key words: Polynomial-like iterative equation, Leading Coefficient Problem, Polynomial solutions, Algebraic variety, Minimal irreducible decomposition

中图分类号:

O193

余志恒,龚小兵. 多项式型迭代方程的多项式解[J]. 数学物理学报, 2019, 39(6): 1352-1364.

Zhiheng Yu,Xiaobing Gong. Polynomial Solutions of the Polynomial-Like Iterative Equation[J]. Acta mathematica scientia,Series A, 2019, 39(6): 1352-1364.

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