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arXiv · 2607.17349

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

Tao Zheng·Hao yuan
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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.

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Tao Zheng, Hao yuan. 2026-07-19. Computing Bunches of Semi-Periodic Solutions of Bivariate Exponential-Trigonometric Polynomial Equations with Separated Variables. https://arxiv.org/abs/2607.17349

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