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I have a list of rational equations and I would like to convert it to a list of polynomial equations. I know that none of the variables and none of the denominators could ever be 0. So far I have tried:

removeDenom[ a_ == b_ ] := Numerator[a]Denominator[b] == Numerator[b]Denominator[a];

ratToPol[eqn_] := removeDenom @* Together @ eqn;

Which seems to do the trick for rational equations with small exponents of the variables but for larger ones, e.g. 

x[1]^24566482 x[2]^25894864 x[3]^36056313 x[5]^2 x[6]^2==(x[1]^10050244 x[2]^10878704 x[3]^15147675 x[6]^8)/x[5]^4+x[1]^39374641 x[2]^41453905 x[3]^57720898 x[5]^4 x[6]^2 x[7]^2

I get the error General::lrgexp Exponent is out of bounds for function 1.

Any ideas on how to proceed? It would be nice if there was a way without using Simplify or other higher-level functions for which it is unknown how they operate exactly.

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    $\begingroup$ Could use Numerator[Together[expressionList]].though this too will balk at large exponents. $\endgroup$
    – Daniel Lichtblau
    Commented Feb 23, 2020 at 16:28
  • $\begingroup$ How about 0 == (eqn /. Equal -> Subtract /. x[i_]^(j_ /; j > 0) -> x[i, j] // Together // Numerator) /. x[i_, j_] -> x[i]^j ? $\endgroup$
    – Akku14
    Commented Oct 20, 2020 at 10:33

2 Answers 2

Reset to default
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Clear["Global`*"]

ratToPol[eqn_Equal] := Module[
  {prod = Times @@ Cases[eqn, t_ :> Denominator[t], 2]},
  Assuming[prod != 0, MultiplySides[eqn, prod] //
    Simplify]]

eqn = x[1]^24566482 x[2]^25894864 x[3]^36056313 x[5]^2 x[
      6]^2 ==
   (x[1]^10050244 x[2]^10878704 x[3]^15147675 x[6]^8)/
     x[5]^4 + 
    x[1]^39374641 x[2]^41453905 x[3]^57720898 x[5]^4 x[6]^2 x[7]^2;

Format[x[n_]] := Subscript[x, n]

eqn2 = ratToPol@eqn

enter image description here

Simplify[Or @@ Thread[(List @@ eqn2[[1]]) == 0]]

enter image description here

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This function works for the example expression

removeDenom[a_ == b_] := Block[{
   u = Times @@ Denominator /@ a,
   v = Times @@ Denominator /@ b},
  Expand[u a v] == Expand[u b v]
  ]

It multiplies the LHS by all the denominators it finds on the RHS and vice versa.

The polynomialGCD error occurs in Together, so don't use ratToPol. Just use removeDenom[p]

Also, removeDenom does NOT play nice with subscripted variables u and v, but it seems to work with x.

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  • $\begingroup$ Expand has the same problems. Also the code doesn't work: removeDenom[x + y - z + 1 == 1/x] returns 4+4x+4y-4z == 4/x :/ This because mapping Denominator on a sum gives the sum of the denominators. An alternative implementation using List@@ would be removedenom[a_ == b_] := With[{eq = a - b}, #[[1]]==#[[2]]&[ Times @@ Denominator[List @@ (eq + 1)]*# & /@ eq ] ] but then the result is unexpanded and indeed: Expand returns Expand::lrgexp: Exponent is out of bounds for function Expand. (although not consistently...) $\endgroup$
    – Gert
    Commented Oct 28, 2019 at 12:41

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