AI 中文总结
研究具有分离变量的二元指数 - 三角多项式方程的求解，通过发展解析代数指数多项式理论，在非退化条件下确定解的分布，由有限个递增且趋于无穷的曲线组成半周期解束，并实现有效算法找曲线及计算半周期根束。
AI 中文摘要
具有分离变量的二元指数 - 三角多项式（BETP）方程形式为\(g(x, e^x, y, \sin y, \cos y) = 0\)，其中\(g\)为多项式，\(x,y\)为实变量。求解此类方程在工程中有用。此外，动态系统中常见的有理系数混合三角多项式和指数多项式的复根计算问题，可归结为求解含两个BETP方程的系统。本文发展了解析代数指数多项式理论，表明在某些非退化条件下，该系统在\(\{(x, y)| x>N, y>M \}\)区域的解位于有限多个递增且趋于无穷的解析代数指数多项式曲线上，这些解由有限多个半周期解束组成，且每束沿特定曲线分布。最后实现了有效算法来找到这些曲线并计算半周期根束。
英文摘要
A bivariate exponential-trigonometric polynomial (BETP) equation with separated variables is of the form g(x, e^x, y, sin y, cos y) = 0 with g a polynomial and x, y real variables. Solving BETP equations with separated variables is useful in engineering. Besides, the problems of computing complex roots of rational-coefficient mixed-trigonometric polynomials and exponential polynomials, which occur frequently in dynamic systems, can both be reduced to solving a system containing two BETP equations with separated variables: g(x, e^x, y, sin y, cos y) = 0 h(x, e^x, y, sin y, cos y) = 0 In this paper, the theory of the analytic algebraic exponential polynomials is developed. Based on which we show that if some non-degenerate conditions hold for the system above, then there are N>0 and M>0 such that all solutions of that system in the quarter {(x, y)| x>N, y>M } lie on the curves of finitely many analytic algebraic exponential polynomials which are increasing and tend to infinity. These solutions consist of finitely many bunches of so-called semi-periodic solutions, and each bunch is entirely distributed along a certain curve. Finally, effective algorithms have been implemented to find those curves and to count those bunches of semi-periodic roots.
