Rigorous analysis of the unexpectedly small solution counts in polynomial systems from Lie-derivative-based parameter estimation

Establish a rigorous analysis explaining why the polynomial systems arising from the differential algebra and rational interpolation parameter estimation approach—constructed using Lie derivatives of outputs of rational ODE models—often have a much lower number of solutions than degree-based expectations, and determine the conditions under which such reductions occur.

Background

A central step in the differential-algebraic parameter estimation framework is solving polynomial systems derived from Lie derivatives of the model outputs. In a toy example, a system of four quadratic equations in four variables has only two solutions despite a naive expectation of sixteen, illustrating a broader empirical observation that such systems often have far fewer solutions than generic bounds suggest.

The authors note that this reduced solution count appears repeatedly across models but currently lacks a rigorous theoretical explanation, leaving a formal analysis as future work.