In this paper we introduce a general class of problems originating from gearbox vibration analysis. Based on a previous work where demodulation was formulated as a matrix approximation problem, we study the specific case applicable to amplitude and phase demodulation. This problem can be rewritten as a polynomial system. Based on algebraic methods such as linear algebra and homological algebra, we focus on the characterization of the problem and solve it in the noise-free case. Copyright (C) 2019. The Authors. Published by Elsevier Ltd. All rights reserved.