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Systems of algebraic equations

  1. Dec 8, 2008 #1
    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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  3. Dec 8, 2008 #2

    HallsofIvy

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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!
     
  4. Dec 8, 2008 #3
    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?
     
  5. Dec 8, 2008 #4

    mathman

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    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.
     
  6. Dec 8, 2008 #5
    So without finding any of these solutions, is there any way to show that they actually exist?
     
  7. Dec 9, 2008 #6

    mathman

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    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.
     
  8. Dec 10, 2008 #7

    HallsofIvy

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    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".
     
  9. Dec 10, 2008 #8

    Office_Shredder

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    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
     
  10. Dec 11, 2008 #9

    mathman

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    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.
     
  11. Dec 12, 2008 #10

    HallsofIvy

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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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