Re: question on a 3 variable polynomial equation

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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 =3D 0.
>
> Let P(a,s,t) =3D (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) =3D 0
>
> We seek rational solutions of P(a,s,t) =3D 0.
>
> I know of 12 such rational solutions:
>
> [a,s,t] =3D [3/34, 2285730/6613289, 337297557/965540194]
> [a,s,t] =3D [3/34, 10355/23766, 397850/922913]
> [a,s,t] =3D [722/699, 10355/23766, 69350/67337]
> [a,s,t] =3D [722/699, 153502520/90640961, 14222008478/19850370459]
> [a,s,t] =3D [4320/5329, 2285730/6613289, 92834190/112425913]
> [a,s,t] =3D [4320/5329, 153502520/90640961, 14970535240/21119343913]
> [a,s,t] =3D [257/19729, 554206077160/5502307479497,
> 4751970732848775/46841143572957961]
> [a,s,t] =3D [257/19729, 809903238990/2823867188761,
> 13669801781456/48005742208937]
> [a,s,t] =3D [896416/1540853, 554206077160/5502307479497,
> 3224681131800/5502307479497]
> [a,s,t] =3D [896416/1540853, 489005742748920/220546618146077,
> 1428720867724168/3749292508483309]
> [a,s,t] =3D [1319445/790789, 809903238990/2823867188761,
> 4678738670640/2823867188761]
> [a,s,t] =3D [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(=C2=B11,=C2=B12,0)

     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,

               Lo=C3=AFc

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<div dir=3D"ltr"><div dir=3D"ltr"><div class=3D"gmail_default" style=3D"fon=
t-family:verdana,sans-serif">On Tue 2 Jun, 2026, at 00:32, American Citizen=
 wrote:</div></div><div class=3D"gmail_quote gmail_quote_container"><blockq=
uote class=3D"gmail_quote" style=3D"margin:0px 0px 0px 0.8ex;border-left:1p=
x solid rgb(204,204,204);padding-left:1ex">This question is about finding m=
ore solutions of a certain 14-degree <br>
polynomial in 3 variables =3D 0.<br>
<br>
Let P(a,s,t) =3D (a^10*s^4 + 2*a^10*s^2*t^2 + a^10*t^4 - a^8*s^6 - <br>
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 + <br>
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 - <br>
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 - <br>
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 + <br=
>
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 + <br=
>
3*s^2*t^4 - t^6) =3D 0<br>
<br>
We seek rational solutions of P(a,s,t) =3D 0.<br>
<br>
I know of 12 such rational solutions:<br>
<br>
[a,s,t] =3D [3/34, 2285730/6613289, 337297557/965540194]<br>
[a,s,t] =3D [3/34, 10355/23766, 397850/922913]<br>
[a,s,t] =3D [722/699, 10355/23766, 69350/67337]<br>
[a,s,t] =3D [722/699, 153502520/90640961, 14222008478/19850370459]<br>
[a,s,t] =3D [4320/5329, 2285730/6613289, 92834190/112425913]<br>
[a,s,t] =3D [4320/5329, 153502520/90640961, 14970535240/21119343913]<br>
[a,s,t] =3D [257/19729, 554206077160/5502307479497, <br>
4751970732848775/46841143572957961]<br>
[a,s,t] =3D [257/19729, 809903238990/2823867188761, <br>
13669801781456/48005742208937]<br>
[a,s,t] =3D [896416/1540853, 554206077160/5502307479497, <br>
3224681131800/5502307479497]<br>
[a,s,t] =3D [896416/1540853, 489005742748920/220546618146077, <br>
1428720867724168/3749292508483309]<br>
[a,s,t] =3D [1319445/790789, 809903238990/2823867188761, <br>
4678738670640/2823867188761]<br>
[a,s,t] =3D [1319445/790789, 489005742748920/220546618146077, <br>
720611087658377235/1877513360277553501]<br>
<br>
How can more rational solutions be found? Is it possible?</blockquote><div>=
<br></div><div style=3D"font-family:verdana,sans-serif" class=3D"gmail_defa=
ult">=C2=A0 =C2=A0 I tried to look for double solutions, so I tried to fact=
or the discriminant</div><div style=3D"font-family:verdana,sans-serif" clas=
s=3D"gmail_default">=C2=A0 in a, s and t. I found the following families of=
 solutions</div><div style=3D"font-family:verdana,sans-serif" class=3D"gmai=
l_default">P(a,a+1/a,0)<br>P(a,0,0)<br>P(a,0,a)<br>P(a,0,-a)<br>P(0,s,s)<br=
>P(0,s,-s)<br>P(=C2=B11,=C2=B12,0)</div><div style=3D"font-family:verdana,s=
ans-serif" class=3D"gmail_default"><br></div><div style=3D"font-family:verd=
ana,sans-serif" class=3D"gmail_default">=C2=A0 =C2=A0 =C2=A0I have attached=
 a file which more or less accounts for my computations.</div><div style=3D=
"font-family:verdana,sans-serif" class=3D"gmail_default">=C2=A0 The comment=
 after a command shows the result. You&#39;ll notice that I read</div><div =
style=3D"font-family:verdana,sans-serif" class=3D"gmail_default">=C2=A0 fro=
m the bottom up.</div><div style=3D"font-family:verdana,sans-serif" class=
=3D"gmail_default"><br></div><div style=3D"font-family:verdana,sans-serif" =
class=3D"gmail_default">=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0Hope this helps,<=
/div><div style=3D"font-family:verdana,sans-serif" class=3D"gmail_default">=
<br></div><div style=3D"font-family:verdana,sans-serif" class=3D"gmail_defa=
ult">=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0Lo=C3=AFc</div>=
</div></div>

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