Eliminating a variable with GroebnerBasis

I have the following two hefty equations:

eqn1 = 110592 u^(14) - 608256 u^(12) - 147456 u^(11) y +
1057536 u^(10) + 602112 u^9*y + 1296 u^8 x^2 + 49152 u^8 y^2 -
591872 u^8 - 9216 u^7 x - 557056 u^7 y + 8856 u^6 x^2 -
131072 u^6 y^2 + 16384 u^6 - 15360 u^5 x - 8487 u^4 x^2 -
57344 u^4 + 139072 u^3 x - 80688 u^2 x^2 + 50176 u^2 -
146944 u x + 107584 x^2 == 0;
eqn2 = 26873856 u^(16) y^4 - 2654208 u^(16) y^2 +
367276032 u^(14) y^4 + 65536 u^(16) + 275816448 u^(14) y^2 +
902886912 u^(12) y^4 + 15592448 u^(14) + 1105416576 u^(12) y^2 -
5751440640 u^(10) y^4 + 61000192 u^(12) - 8064265152 u^(10) y^2 -
17252143344 u^8 y^4 + 668471424 u^(10) - 12938192552 u^8 y^2 +
52402014720 u^6 y^4 + 1938827977 u^8 + 79391817472 u^6 y^2 +
74950760448 u^4 y^4 - 14469805664 u^6 - 73272450048 u^4 y^2 -
277783609344 u^2 y^4 + 8493420928 u^4 + 20904001536 u^2 y^2 +
185189072896 y^4 + 3487442944 u^2 - 29359243264 y^2 + 282300416 ==
0;


Please note that x does not appear in eqn2. I want to eliminate u. I tried (in addition to Eliminate[]):

GroebnerBasis[{eqn1, eqn2}, {x, y}, {u}]


including various combinations of the options: MonomialOrder, Method and Sort. These just run all night, without producing anything.

I got a tad optimistic when I invoked GroebnerBasis[] without eliminating u, and it came back almost immediately with a huge result of 43 equations and gargantuan coefficients. I then tried to work with just two equations at a time -- nada.

Any ideas how to approach this differently?

When does one know that there is no answer, vs. Mathematica will not come up with an answer, vs. to let it continue to chomp away?

Thank you.

• "a huge result of 43 equations and gargantuan coefficients" - I would venture that this is part of the reason the elimination of u is taking quite a while. Feb 17, 2021 at 7:17
• If I am not mistaken, you have 2 equations of the variables: x,y,u. To eliminate u, it takes one equation. This leaves one equation of two variables. But you can not solve a single equation for two variables. Feb 17, 2021 at 8:59

How about using Resultant? The result is very large so not posted (noticed I removed the =0 from both expressions):

eqn1 = 110592 u^(14) - 608256 u^(12) - 147456 u^(11) y +
1057536 u^(10) + 602112 u^9*y + 1296 u^8 x^2 + 49152 u^8 y^2 -
591872 u^8 - 9216 u^7 x - 557056 u^7 y + 8856 u^6 x^2 -
131072 u^6 y^2 + 16384 u^6 - 15360 u^5 x - 8487 u^4 x^2 -
57344 u^4 + 139072 u^3 x - 80688 u^2 x^2 + 50176 u^2 -
146944 u x + 107584 x^2;
eqn2 = 26873856 u^(16) y^4 - 2654208 u^(16) y^2 +
367276032 u^(14) y^4 + 65536 u^(16) + 275816448 u^(14) y^2 +
902886912 u^(12) y^4 + 15592448 u^(14) + 1105416576 u^(12) y^2 -
5751440640 u^(10) y^4 + 61000192 u^(12) - 8064265152 u^(10) y^2 -
17252143344 u^8 y^4 + 668471424 u^(10) - 12938192552 u^8 y^2 +
52402014720 u^6 y^4 + 1938827977 u^8 + 79391817472 u^6 y^2 +
74950760448 u^4 y^4 - 14469805664 u^6 - 73272450048 u^4 y^2 -
277783609344 u^2 y^4 + 8493420928 u^4 + 20904001536 u^2 y^2 +
185189072896 y^4 + 3487442944 u^2 - 29359243264 y^2 + 282300416;

Resultant[eqn1, eqn2, u]

• Thank you, I thought that Resultant was using the same backend as GroebnerBasis, but apparently they are different enough. Feb 18, 2021 at 20:29