Systems of algebraic equations

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SUMMARY

This discussion focuses on the existence of solutions for systems of nonlinear algebraic equations. It establishes that while linear systems can often be analyzed based on the number of equations relative to unknowns, this criterion does not directly apply to nonlinear systems. For instance, a single nonlinear equation can have multiple solutions or none at all, complicating the determination of solution existence in a system. The Inverse Function Theorem is mentioned as a tool that provides conditions under which solutions may exist, but it does not offer a universal method applicable to all scenarios.

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  • Understanding of linear and nonlinear algebraic equations
  • Familiarity with the Inverse Function Theorem
  • Knowledge of solution existence criteria in algebraic systems
  • Basic concepts of differentiability in mathematical functions
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  • Research the Inverse Function Theorem and its applications in nonlinear equations
  • Explore methods for determining solution existence in nonlinear systems
  • Study examples of nonlinear equations with multiple solutions
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Mathematicians, students studying algebraic systems, and researchers interested in nonlinear equations and their solution properties.

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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)
 
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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!
 
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?
 
jimmy1 said:
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.
 
mathman said:
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?
 
jimmy1 said:
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.
 
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".
 
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
 
HallsofIvy said:
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.
 
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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!"
 

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