Is there a general procedure to detect if in a system of m-nonlinear equations (also non polynomial) of n-unknowns some of the equations are redundant? Can the rank of the Jacobian matrix tell me something? The systems I handle is usually underdetermined, so m is less or equal to n.
How I numerically solve the system: I use the iterative Gauss-Newton method. Since there are more unknowns than equations, I solve the linear equation involving the Jacobian matrix in the least squares sense. I was thinking that if the rank of the Jacbian matrix is not maximum at the solution point, then there must be some redundant equations. Actually the pair of equations eq 1: $y_1 - y_2 = 10$ eq 2: $(x_1-x_2)^2+(y_1-y_2)^2=100$ with unknowns $x_1,x_2,y_1,y_2$ is non-redundant, but the rank of the Jacobian matrix is 1 in the subset where $x_1=x_2$. Maybe evaluating the Jacobian Rank only at one point is not enough?
I found a similar discussion here https://www.physicsforums.com/threads/how-to-detect-redundant-equation-from-a-system-of-nonlinear-equation.418848/