Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Question

Consider the symmetric polynomial

$$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$

How can we write $P$ as a polynomial in the variables $st,t+s$?

The Fundamental theorem of symmetric polynomials implies this is possible, but I am having trouble doing it in practice.

Answer 1

$P(s,t)=s^4+t^4-2s^2-2t^2+2$.

Now, $s^2+t^2=\sigma^2-2\pi$, where $\sigma$ and $\pi$ are the sum and product of $s$ and $t$. Thus, $s^4+t^4=(s^2+t^2)^2-2s^2t^2=(\sigma^2-2\pi)^2-2\pi^2=\sigma^4-4\pi\sigma^2+2\pi^2$.

So $P(s,t)=2-2\sigma^2+4\pi+\sigma^4-4\pi\sigma^2+2\pi^2$.

Answer 2

\begin{align} & P(s, t) = (s^2 - 1 + t^2 - 1)^2 - 2(s^2 - 1)(t^2 - 1) \\ = & ((s + t)^2 - 2st - 2)^2 - 2[s^2t^2 - (s^2 + t^2) + 1] \\ = & ((s + t)^2 - 2st - 2)^2 - 2[s^2t^2 - (s + t)^2 + 2st + 1] \end{align}

Answer 3

Hint

$$s^4+t^4=(s^2+t^2)^2-2(st)^2$$ and $$s^2+t^2=(s+t)^2-2(st).$$

Also see that $$(s^2-1)^2+(t^2-1)^2=s^4+t^4-2(s^2+t^2)+2$$