Algorithm 1: Probabilistic algorithm to test local algebraic observability in polynomial time

Preprocesing Construct a straight-line program encoding the variational system $\nabla P$ with $P=\dot{\tilde{x}}-f(u\left(t\right),\tilde{x}\left(t\right))$ and the expressions used during its integration. Specialization Specialisation of the parameters, ${\theta }^{*}$, and the inputs, $u^{*}$ Power Series Solution Computation of the power series solution of $\nabla P$ at order $n_{\tilde{x}}+1$ with a specialised value for the states Jacobian computation Evaluation of $\nabla y$ on the previous results, giving the coefficients of the Jacobian matrix Rank computation Calculation of the matrix rank and transcendence degree if transcendence degree = 0 then  |    System is algebraically observable else  |    Determine which variable or variables are not observable. end