The command IteratedResultant(p, rc, R) returns the iterated resultant of p with respect to rc.

If p is a constant or all variables in p are free with respect to rc, then p is returned. Otherwise, if v is the largest variable of p algebraic with respect to rc, then IteratedResultant(r, Under(v, rc, R), R) is returned, where r is the resultant of p and the polynomial $\mathrm{Polynomial}\left(v\,\mathrm{rc}\,R\right)$.

This command is part of the RegularChains[ChainTools] package, so it can be used in the form IteratedResultant(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IteratedResultant](..).

The commands IteratedResultantDim0 and IteratedResultantDim1 provide fast algorithms for computing iterated resultants in prime characteristic and with regular chains in dimensions $0$ and $1$ respectively.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,t\,s\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[x+y^2-t\,t^2-s\right]$

$F≔\left[y^2-t+x\,t^2-s\right]$

$p≔xyt$

$p≔xyt$

$\mathrm{dec}≔\mathrm{Triangularize}\left(F\,R\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{IteratedResultant}\left(p\,\mathrm{dec}\left[1\right]\,R\right)$

$-s{y}^6+s^2{y}^2$