Eliminate G variable from this system of two equations?

Questions
,
1

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]
2

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

3

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

4

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.

5

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.