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Eliminate G variable from this system of two equations?

Posted 3 months ago
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Consider the following code:

Eliminate[{-A^16 + 3 A^11 F - 3 A^6 F^2 + A F^3 + 10 A^14 G - 
    20 A^9 F G + 10 A^4 F^2 G - 40 A^12 G^2 + 55 A^7 F G^2 - 
    15 A^2 F^2 G^2 + 80 A^10 G^3 - 85 A^5 F G^3 + 5 F^2 G^3 - 
    75 A^8 G^4 + 75 A^3 F G^4 - 25 A F G^5 + 75 A^4 G^6 - 
    75 A^2 G^7 + 25 G^8 + 25 A^6 H - 75 A^4 G H + 75 A^2 G^2 H - 
    25 G^3 H - 15 A^12 R + 5 A^7 F R + 10 A^2 F^2 R + 105 A^10 G R + 
    15 A^5 F G R + 5 F^2 G R - 300 A^8 G^2 R - 75 A^3 F G^2 R + 
    475 A^6 G^3 R + 25 A F G^3 R - 500 A^4 G^4 R + 375 A^2 G^5 R - 
    125 G^6 R - 25 A^8 R^2 + 25 A^3 F R^2 + 75 A^6 G R^2 + 
    50 A F G R^2 - 250 A^2 G^3 R^2 + 125 G^4 R^2 + 125 A^2 G R^3 == 
   0 , -A^25 - 3125 A^10 B + 5 A^20 F - 10 A^15 F^2 + 10 A^10 F^3 - 
    5 A^5 F^4 + F^5 + 25 A^23 G + 15625 A^8 B G - 100 A^18 F G + 
    150 A^13 F^2 G - 100 A^8 F^3 G + 25 A^3 F^4 G - 275 A^21 G^2 - 
    31250 A^6 B G^2 + 850 A^16 F G^2 - 900 A^11 F^2 G^2 + 
    350 A^6 F^3 G^2 - 25 A F^4 G^2 + 1750 A^19 G^3 + 
    31250 A^4 B G^3 - 4000 A^14 F G^3 + 2750 A^9 F^2 G^3 - 
    500 A^4 F^3 G^3 - 7125 A^17 G^4 - 15625 A^2 B G^4 + 
    11375 A^12 F G^4 - 4500 A^7 F^2 G^4 + 250 A^2 F^3 G^4 + 
    19375 A^15 G^5 + 3125 B G^5 - 20000 A^10 F G^5 + 
    3750 A^5 F^2 G^5 - 35625 A^13 G^6 + 21250 A^8 F G^6 - 
    1250 A^3 F^2 G^6 + 43750 A^11 G^7 - 12500 A^6 F G^7 - 
    34375 A^9 G^8 + 3125 A^4 F G^8 + 15625 A^7 G^9 - 3125 A^5 G^10 + 
    25 A^21 R - 100 A^16 F R + 150 A^11 F^2 R - 100 A^6 F^3 R + 
    25 A F^4 R - 375 A^19 G R + 1125 A^14 F G R - 1125 A^9 F^2 G R + 
    375 A^4 F^3 G R + 2125 A^17 G^2 R - 4500 A^12 F G^2 R + 
    2625 A^7 F^2 G^2 R - 250 A^2 F^3 G^2 R - 4875 A^15 G^3 R + 
    6500 A^10 F G^3 R - 1500 A^5 F^2 G^3 R - 125 F^3 G^3 R - 
    1875 A^13 G^4 R + 3750 A^8 F G^4 R - 1875 A^3 F^2 G^4 R + 
    36250 A^11 G^5 R - 22500 A^6 F G^5 R + 1875 A F^2 G^5 R - 
    87500 A^9 G^6 R + 25000 A^4 F G^6 R + 103125 A^7 G^7 R - 
    9375 A^2 F G^7 R - 62500 A^5 G^8 R + 15625 A^3 G^9 R + 
    375 A^17 R^2 - 500 A^12 F R^2 - 125 A^7 F^2 R^2 + 
    250 A^2 F^3 R^2 - 6250 A^15 G R^2 + 6250 A^10 F G R^2 + 
    39375 A^13 G^2 R^2 - 25625 A^8 F G^2 R^2 + 1875 A^3 F^2 G^2 R^2 - 
    124375 A^11 G^3 R^2 + 48750 A^6 F G^3 R^2 - 2500 A F^2 G^3 R^2 + 
    215625 A^9 G^4 R^2 - 43750 A^4 F G^4 R^2 - 200000 A^7 G^5 R^2 + 
    12500 A^2 F G^5 R^2 + 75000 A^5 G^6 R^2 + 3125 F G^6 R^2 + 
    15625 A^3 G^7 R^2 - 15625 A G^8 R^2 - 1875 A^13 R^3 + 
    625 A^8 F R^3 + 1250 A^3 F^2 R^3 + 3125 A^11 G R^3 - 
    3125 A^6 F G R^3 + 25000 A^9 G^2 R^3 + 6250 A^4 F G^2 R^3 - 
    106250 A^7 G^3 R^3 - 3125 A^2 F G^3 R^3 + 175000 A^5 G^4 R^3 - 
    3125 F G^4 R^3 - 140625 A^3 G^5 R^3 + 46875 A G^6 R^3 - 
    3125 A^9 R^4 + 3125 A^4 F R^4 + 15625 A^7 G R^4 - 
    31250 A^5 G^2 R^4 + 31250 A^3 G^3 R^4 - 15625 A G^4 R^4 + 
    3125 A^5 R^5 == 0 }, G]
4 Replies

The last equation is of degree 1 with respect to G. You can solve it and replace in the rest of the system.

Posted 3 months ago

Hi, there are two equations , with inconnu ( A , G ) , so , I want , just one equation but , without G. ( R - H - F - B ) = parameter.

Well, on a positive side you can eliminate C, D or E.

Those equations are of degree 9 and 11 with respect to G, so I don't think it is possible. Use CoefficientList to verify the order yourself.

BTW, you should be using lowercase variables names.

The function Resultant is good for this type of problem (though this can be sloooow..). After changing variables to lower case:

Timing[res = Resultant[polys[[1]], polys[[2]], g];]

(* Out[30]= {1918.83, Null} *)

It's a big expression.

In[31]:= LeafCount[res]

(* Out[31]= 321649 *)

It can be factored (`FactorSquareFree is what I would usually do first though, as it is faster and, in this case, gives the same result).

Timing[res2 = FactorList[res];]

(* Out[32]= {29.9531, Null} *)

The first two are probably not useful factors (I checked leaf counts before deciding this).

Most[res2]

(* Out[48]= {{95367431640625, 1}, {a^5 - f - 5 a r, 15}} *)

The remaining factor has squarefree and has leaf count of "only" around 70K. It's the one of interest, in terms of the elimination problem at hand.

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