How to detect redundant equation from a system of nonlinear equation?

[gohkgohkgohk]

How to detect redundant equation from a system of nonlinear equation?
It means how to find out a system of nonlinear equation is "linear independence"?
One equation from the system can not be represented by the others in the system of nonlinear equation.

 

[Eynstone]

Welcome gohkgohkgohk .
I'm afraid there's no general method of spotting a redundant equation from a set. The Jacobian helps a liitle if you have differentiable non-linear equations.

 

[kaige]

Hi Eynstone, could you share more details on how to determine the redundant equation using Jacobian? My suspection is even its Jacobian matrix has redundant rows, the non-linear equation system may not have redundant equations.

 

[algebrat]

I checked the section on functional dependence from Gerald Folland's Advanced Calculus.

There it says, if you consider on open set U, then

Jacobian is zero throughout U if and only if the functions have some (possibly various) functional dependence(s) throughout U.

So it's not enough for the Jacobian to vanish at a single point, calculus needs an open set to draw conclusions. To deal with single points, I think is part of the practicality of the subject algebraic geometry, which more or less deals with systems of polynomials, as far as I know, but looks like a very abstract subject.