The Determinant of an $n\times n$ matrix, viewed as a polynomial in the entries, is irreducible. But when it is restricted to the subspace of alternate matrices, it becomes reducible, actually the square of a polynomial known as the Pfaffian.

Likewise, the Resultant of a pair $(P,Q)$ of polynomials of respective degrees $\le n,m$, when viewed as a polynomial in the coefficients of $P$ and $Q$, is an irreducible polynomial ($n+m+2$ indeterminates, degree $n+m$).

Is there a natural subspace $E$ of $k_n[X]\times k_m[X]$, such that the restriction of the Resultant to $E$, viewed as a polynomial in the coordinates, splits in a non-trivial way ?

*Low temperature example:* Let $n,m$ be even, and $E$ be the space of pairs of even polynomials. Then the restriction of the Resultant over $E$ is a square. If $P(X)=p(X^2)$ and $Q(X)=q(X^2)$, then ${\rm Res}(P,Q)=({\rm Res}(p,q))^2$. This follows from the formula ${\rm Res}(P,Q)=\prod(x_i-y_j)$ in terms of the roots $x_i$ and $y_j$ (to be adapted if $P,Q$ are not monic).

*Less cold example:* Let $n,m$ be even, and $F$ be the space of pairs of reciprocal polynomials. Then the restriction of the Resultant over $F$ is a square. If $P(X)=X^np(X+\frac1X)$ and $Q(X)=X^mq(X+\frac1X)$, then ${\rm Res}(P,Q)=({\rm Res}(p,q))^2$.