Transforming polynomial ODE systems into matrix Ricatti systems

Determine whether a polynomial ordinary differential equation system admits a transformation (for example, via introduction of new variables) into the matrix Ricatti equation dW/dt = A(t) + B(t) W + W C(t) + W D(t) W, where W is an (n × k) matrix and A(t), B(t), C(t), and D(t) are matrices of compatible dimensions.

Background

The paper develops algorithms based on Newton polytopes to verify whether a system of first-order nonlinear ODEs admits a nonlinear superposition rule by checking the finite-dimensional closure of associated polynomial vector fields under the Lie bracket. A central example is the matrix Ricatti equation, which the authors discuss along with a discretization that preserves nonlinear superposition and is strongly consistent.

Prior work shows that polynomial ODE systems can sometimes be transformed into quadratic forms via the introduction of auxiliary variables (monomial quadratization). The open problem asks whether an analogous transformation exists that maps polynomial ODE systems into the matrix Ricatti form, which would enable leveraging known superposition-preserving and strongly consistent numerical schemes for a broader class of systems.