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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:

$$y_1 − y_2 = 10\, ,\\ (x_1 − x_2)^2 + (y_1 − y_2)^2 = 100\, ,$$

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/

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  • $\begingroup$ It is not clear whether you consider the under- or over-determined case, it kind-of oscillates. I would expect redundant equations in the case where there are more equations than variables. But in general there is not just the one easily identifiable superfluous equation. You could remove any equation that is locally dependent. $\endgroup$ Commented Aug 3, 2023 at 9:34
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    $\begingroup$ Yes, you can use such arguments. Linear independence of the derivatives/gradients of some selection of the equations at some random point is highly indicative of functional independence everywhere. The probability increases if the independence of the same selection holds at several random points. This is especially true if the equations are polynomial or analytic. With just smooth functions one can construct (contrived) examples where such heuristic fails. $\endgroup$ Commented Aug 3, 2023 at 10:12
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    $\begingroup$ If the equations are linearized at the root, it should be right to analyze the Jacobian at the root to decide whether the system is over- or under-determined. But in some cases you'll need higher derivatives, say with two equations $y_1=x^2$ and $y_2=x^3$, linearizing them at the root x=0 would not yield anything useful. $\endgroup$ Commented Aug 4, 2023 at 14:05
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    $\begingroup$ It is case dependent, say for the system $y_1=x^{100}$ and $y_2=x^{200}$, you'll need derivatives of 200$^{th}$ order to analyze the local behavior. We'd need to know something about the structure of those equations to say something more specific. $\endgroup$ Commented Aug 4, 2023 at 14:14
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    $\begingroup$ But this system is clearly undetermined, the first equation is $y1-y2=10$ and using the first equation in the second yields $x1=x2$, so there is infinite number of solutions here: $x1=x2=t$, $y2=p$, $y1=p+10$ for any $p,t$. $\endgroup$ Commented Aug 4, 2023 at 14:24

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In the example you give, the two equations are not redundant. Each of the two equations describes a set of lines in the 2d plane, and the lines happen to be tangential at a specific point -- which is expressed as the Jacobian computed at that point being singular. But that doesn't mean that the two lines are globally the same.

I doubt that there is a general technique to show that arbitrary nonlinear equations are redundant. I could imagine that there are techniques that work for polynomial equations, and in that case this is a question in the field of algebraic geometry. You might want to ask people there, or look into the documentation of software packages such as Bertini to see whether they have approaches to deal with the issue.

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