Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

Existing structural analysis methods may fail to identify all hidden constraints in systems of differential-algebraic equations with parameters, particularly when the system is structurally unamenable for certain parameter values. In this paper, the authors address numerical methods for polynomial systems of differential-algebraic equations using numerical real algebraic geometry to resolve such issues. Initially, the authors propose an embedding method that constructs an equivalent system with a full-rank Jacobian matrix for any given real analytic system. Secondly, the authors introduce a witness point method, which assists in detecting the constant rank of a component of the constraints in such systems. Finally, these two methods lead to a comprehensive numerical global structural analysis method for polynomial differential-algebraic equations across all components of constraints.