Re: question on a 3 variable polynomial equation

On Tue 2 Jun, 2026, at 00:32, American Citizen wrote:

This question is about finding more solutions of a certain 14-degree
polynomial in 3 variables = 0.

Let P(a,s,t) = (a^10*s^4 + 2*a^10*s^2*t^2 + a^10*t^4 - a^8*s^6 -
3*a^8*s^4*t^2 + 4*a^8*s^4 - 3*a^8*s^2*t^4 + 4*a^8*s^2*t^2 - a^8*t^6 +
4*a^8*t^4 - 2*a^6*s^6 - 4*a^6*s^4*t^2 + 6*a^6*s^4 - 6*a^6*s^2*t^4 -
4*a^6*t^6 + 6*a^6*t^4 - 2*a^4*s^4*t^2 + 4*a^4*s^4 - 4*a^4*s^2*t^2 -
6*a^4*t^6 + 4*a^4*t^4 - a^2*s^8 + 2*a^2*s^6 - 4*a^2*s^4*t^2 + a^2*s^4 +
6*a^2*s^2*t^4 - 2*a^2*s^2*t^2 - 4*a^2*t^6 + a^2*t^4 + s^6 - 3*s^4*t^2 +
3*s^2*t^4 - t^6) = 0

We seek rational solutions of P(a,s,t) = 0.

I know of 12 such rational solutions:

[a,s,t] = [3/34, 2285730/6613289, 337297557/965540194]
[a,s,t] = [3/34, 10355/23766, 397850/922913]
[a,s,t] = [722/699, 10355/23766, 69350/67337]
[a,s,t] = [722/699, 153502520/90640961, 14222008478/19850370459]
[a,s,t] = [4320/5329, 2285730/6613289, 92834190/112425913]
[a,s,t] = [4320/5329, 153502520/90640961, 14970535240/21119343913]
[a,s,t] = [257/19729, 554206077160/5502307479497,
4751970732848775/46841143572957961]
[a,s,t] = [257/19729, 809903238990/2823867188761,
13669801781456/48005742208937]
[a,s,t] = [896416/1540853, 554206077160/5502307479497,
3224681131800/5502307479497]
[a,s,t] = [896416/1540853, 489005742748920/220546618146077,
1428720867724168/3749292508483309]
[a,s,t] = [1319445/790789, 809903238990/2823867188761,
4678738670640/2823867188761]
[a,s,t] = [1319445/790789, 489005742748920/220546618146077,
720611087658377235/1877513360277553501]

How can more rational solutions be found? Is it possible?

I tried to look for double solutions, so I tried to factor the discriminant

in a, s and t. I found the following families of solutions

P(a,a+1/a,0)
P(a,0,0)
P(a,0,a)
P(a,0,-a)
P(0,s,s)
P(0,s,-s)
P(±1,±2,0)

I have attached a file which more or less accounts for my computations.

The comment after a command shows the result. You'll notice that I read

from the bottom up.

Hope this helps,

Loïc