Systems of algebraic equations

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Discussion Overview

The discussion centers on the existence of solutions for systems of algebraic equations, particularly focusing on the transition from linear to nonlinear equations. Participants explore whether the criteria for determining solution existence in linear systems apply to nonlinear systems, and they seek general methods for establishing the existence of solutions in nonlinear contexts.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation

Main Points Raised

  • Some participants note that the relationship between the number of equations and unknowns in linear systems does not necessarily apply to nonlinear systems, citing examples of single nonlinear equations with varying numbers of solutions.
  • There is a question about whether a general method exists to demonstrate the existence of solutions for systems of nonlinear equations.
  • One participant suggests that having more equations than unknowns generally leads to no solutions, but acknowledges that dependencies among equations could allow for solutions.
  • Another participant mentions the Inverse Function Theorem as a condition for the existence of solutions under certain differentiability assumptions, but notes it does not provide a comprehensive solution method.
  • Clarifications are made regarding the interpretation of the original question, emphasizing the focus on nonlinear equations rather than the existence of multiple solutions for single equations.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of linear criteria to nonlinear systems, and there is no consensus on a general method for proving the existence of solutions in nonlinear contexts. The discussion remains unresolved regarding the best approaches to this problem.

Contextual Notes

Limitations include the lack of a general approach applicable to all nonlinear systems and the dependence on specific conditions such as differentiability for certain theorems mentioned.

jimmy1
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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.
 
  • #10
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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