Systems of algebraic equations

[jimmy1]

If I had a linear system of algebraic equations, then I can relate the number of unknowns to the number of equations to determine if a solution exists. However, does this criteria carry over to nonlinear equations?

For example, I have a set of m>2 non linear equations and I have 2 unknowns. In general, is it possible to get a solution to this or even just show a solution exists? (I know if they were linear then no solution exists, but not sure with nonlinear eqns)

 

[HallsofIvy]

No it doesn't as, for example, the single equation x²= 4 in one "unknown" has two solutions while the single equation x²= 0 has exactly one solution and the single equation x²= -4 has no solutions. If it doesn't work for one equation it surely won't work for a system of equations!

 

[jimmy1]

Ok, so is there any general method to show that a solution exists (or dosn't exist) to a system of non linear equations? What would the best approach be?

 

[mathman]

If I had a linear system of algebraic equations, then I can relate the number of unknowns to the number of equations to determine if a solution exists. However, does this criteria carry over to nonlinear equations?

For example, I have a set of m>2 non linear equations and I have 2 unknowns. In general, is it possible to get a solution to this or even just show a solution exists? (I know if they were linear then no solution exists, but not sure with nonlinear eqns)

If you have more equations than unknowns, there will in gnereal be no solutions. However, in some situations, the equations are not independent so you may have solutions. For example with 2 unknowns and more than 2 equations, try finding solutions using 2 equations and check to see if they satisfy the other equations.

jimmy1's comment is to a different question. When equations are non-linear, there may be mutiple solutions.

 

[jimmy1]

When equations are non-linear, there may be mutiple solutions.

So without finding any of these solutions, is there any way to show that they actually exist?

 

[mathman]

So without finding any of these solutions, is there any way to show that they actually exist?

In some cases, yes. However, there is no general approach that will work in all situations.

Correction in my previous note - I meant Halls of Ivy comment, not jimmy1.

 

[HallsofIvy]

mathman, in the original post, jimmy1 specifically asked about "non-linear" equations. I don't know why you said my comment "is to a different question".

 

[Office_Shredder]

The Inverse Function Theorem gives conditions on when a set of equations has a solution (assuming the equations are sufficiently differentiable) but doesn't tell you anything else

 

[mathman]

mathman, in the original post, jimmy1 specifically asked about "non-linear" equations. I don't know why you said my comment "is to a different question".

My interpretation of his original question has is whether or not a system of several equations (m) in various unknowns (n), where n < m can be solved. I don't think he had any concern about multiple solutions to a single equation in one unknown.

 

[HallsofIvy]

And that was what I responded to- see my last sentence, " If it doesn't work for one equation it surely won't work for a system of equations!"