Let, P ( x ) Q ( x ) and M ( x ) N ( x ) be two rational functions consisting of polynominals P ( x ) , Q ( x ) , M ( x ) , N ( x ) . The polyniminals are defined with respect to variable x. Now, if a rational function is formed by summation of the rational functions, does there exist a general way to evaluate the zeros of P ( x ) Q ( x ) + M ( x ) N ( x ) ? If yes, where will the zeros of P ( x ) Q ( x ) and M ( x ) N ( x ) map to?Edit 1: I know that, roots of P ( x ) N ( x ) will be union of the roots of P(x) and N(x), and so on so forth for Q ( x ) M ( x ). So, few possible roots can be calculated by taking the intersection of the roots of P ( x ) N ( x ) and Q ( x ) M ( x ). But I am not interested in them, I want solution in general.Edit 2: What if, P ( x ) Q ( x ) is a Pade approximant of order (m,n)?

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Let,             P      (      x      )              Q      (      x      )       and             M      (      x      )              N      (      x      )       be two rational functions consisting of polynominals       P    (    x    )    ,      Q    (    x    )    ,      M    (    x    )    ,      N    (    x    )  . The polyniminals are defined with respect to variable x. Now, if a rational function is formed by summation of the rational functions, does there exist a general way to evaluate the zeros of             P      (      x      )              Q      (      x      )        +            M      (      x      )              N      (      x      )      ? If yes, where will the zeros of             P      (      x      )              Q      (      x      )       and             M      (      x      )              N      (      x      )       map to?Edit 1: I know that, roots of   P  (  x  )  N  (  x  ) will be union of the roots of P(x) and N(x), and so on so forth for   Q  (  x  )  M  (  x  ). So, few possible roots can be calculated by taking the intersection of the roots of   P  (  x  )  N  (  x  ) and   Q  (  x  )  M  (  x  ). But I am not interested in them, I want solution in general.Edit 2: What if,             P      (      x      )              Q      (      x      )       is a Pade approximant of order (m,n)?

tilsjaskak6 · Accepted Answer

Hint:      P    Q    +      M    N    =            P      N      +      Q      M              Q      N      so, if we know the roots of   P  (  x  )  ,  Q  (  x  )  ,  M  (  x  )  ,  N  (  x  ), it is easy to find the roots of the denominator, but not ( in general) the roots of the numerator.

grenivkah3z · Accepted Answer

There is no particular relation between the roots of two functions and their sum. Think of   f  (  x  )  +  f  (  x  ), or   f  (  x  )  −  f  (  x  ), or   f  (  x  )  +  1.Furthermore, if   f and   g are polynomials only known by their roots, they are indeterminate to a constant factor and the roots of   λ  f  +  μ  g can be anything.

Let, P ( x ) </mrow> Q ( x

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