Computing Bunches of Semi-Periodic Solutions of Bivariate Exponential-Trigonometric Polynomial Equations with Separated Variables

2026-07-19

AI Summary

This study investigates a class of bivariate exponential-trigonometric polynomial systems with separated variables, which commonly arise in dynamical systems and engineering applications. By advancing the theory of analytic algebraic exponential polynomials, the work establishes that under non-degeneracy conditions, all solutions in the far region of the first quadrant can be reduced to clusters of semi-periodic solutions lying on finitely many analytic curves that monotonically tend to infinity. The research innovatively transforms the problem of complex root finding into an analysis of the geometric structure of solution clusters, thereby developing a distribution theory for solutions of such systems. Furthermore, it devises an efficient algorithm based on a symbolic-numeric hybrid approach to accurately locate solution curves and count solution clusters, offering a novel computational framework for solving mixed exponential-trigonometric systems.

Abstract

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.

Problem

• bivariate exponential-trigonometric polynomial
• separated variables
• semi-periodic solutions
• analytic algebraic exponential polynomials
• root finding

Innovation

• analytic algebraic exponential polynomials
• semi-periodic solutions
• bivariate exponential-trigonometric polynomial
• separated variables
• effective algorithms