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
}

\\ Double solutions in t
fdt = factor(poldisc(P(a,s,t),t))
\\ [                             s 12]
\\ [                             a  8]
\\ [                       a^2 + 1 20]
\\ [                 a^2 - s*a + 1  1]
\\ [                 a^2 + s*a + 1  1]
\\ [       a^6 + 2*a^4 + a^2 + s^2  1]
\\ [   2*a^6 + 4*a^4 + 2*a^2 + s^4  4]
\\ [4*a^6 + 8*a^4 + 4*a^2 + 27*s^4  2]
factor(P(a,a+1/a,t))
\\ [                                                                a -4]
\\ [                                                          a^2 + 1  4]
\\ [                                                                t  2]
\\ [a^4*t^4 + (2*a^6 - 3*a^2)*t^2 + (a^8 + 4*a^6 + 4*a^4 + 4*a^2 + 3)  1]

P(a,a+1/a,t)==P(a,-a-1/a,t)
\\ 1
P(a,a+1/a,0)
\\ 0

factor(P(a,0,t))
\\ [a^2 + 1 4]
\\ [      t 4]
\\ [  t - a 1]
\\ [  t + a 1]
P(a,0,0)
\\ 0
P(a,0,a)
\\ 0
P(a,0,-a)
\\ 0

factor(P(0,s,t))
\\ [t - s 3]
\\ [t + s 3]
P(0,s,s)
\\ 0
P(0,s,-s)
\\ 0

\\ Double solutions in s
fds = factor(poldisc(P(a,s,t),s))
[                                a 10]
[                          a^2 + 1 28]
[                                t 12]
[                            t - a  1]
[                            t + a  1]
[t^4 - 4*a^2*t^2 + (2*a^6 + 2*a^2)  4]
[complicated                        2]

\\ I did not really try to handle the complicated polynomial

factor(P(a,s,a))
\\ [s           2]
\\ [complicated 1]
P(a,0,a)
\\ 0
\\ I don't know how to handle the complicated polynomial.

P(a,s,-a)==P(a,s,a)
\\ 1

P(a,s,0)/s^4
\\ a^10 + (-s^2 + 4)*a^8 + (-2*s^2 + 6)*a^6 + 4*a^4 + (-s^4 + 2*s^2 + 1)*a^2 + s^2
factor(poldisc(P(a,s,0)/s^4)/s^26)
\\ [     s - 2 2]
\\ [     s + 2 2]
\\ [   s^2 - 2 8]
\\ [27*s^2 - 4 2]
factor(P(a,2,0))
\\ [                a - 1 2]
\\ [                a + 1 2]
\\ [a^6 + 2*a^4 + a^2 + 4 1]
P(1,2,0)
\\ 0
P(-1,2,0)
\\ 0
\\ The nearly rational solutions (s=\sqrt{2} and s=2/(3\sqrt{3}) do not give
\\ rise to real solutions)

\\ Double solutions in a
fda = factor(poldisc(P(a,s,t),a))
[                                                                      s 48]
[                                                                  t - s  3]
[                                                                  t + s  3]
[                                                              t^2 + s^2  2]
[t^6 + (3*s^2 + 4)*t^4 + (3*s^4 - 4*s^2 - 4)*t^2 + (s^6 - 4*s^4 + 4*s^2)  4]
[complicated                                                              2]

factor(P(a,s,s))
\\ [                                                                      s 4]
\\ [                                                                      a 2]
\\ [4*a^8 + (-8*s^2 + 12)*a^6 + (-16*s^2 + 12)*a^4 + (-8*s^2 + 4)*a^2 - s^4 1]
P(0,s,s)
\\ 0
P(a,0,0)
\\ 0

\\ P(a,0,t) already treated above

\\ I don't know how to handle the more complicated polynomials of fda