The function call Tail(p, R) returns the tail of p with respect to the variable ordering of R, that is the reductum of p regarded as a univariate polynomial in its main variable.

It is assumed that p is a nonconstant polynomial.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$p≔\left(y+1\right)x^3+\left(z+4\right)x+3$

$p≔\left(y+1\right)x^3+\left(z+4\right)x+3$

$\mathrm{MainVariable}\left(p\,R\right)$

$x$

$\mathrm{Initial}\left(p\,R\right)$

$y+1$

$\mathrm{MainDegree}\left(p\,R\right)$

$3$

$\mathrm{Rank}\left(p\,R\right)$

$x^3$

$\mathrm{Tail}\left(p\,R\right)$

$xz+4x+3$

$\mathrm{Separant}\left(p\,R\right)$

$3x^{2}y+3x^2+z+4$

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

$R≔\mathrm{polynomial\_ring}$

$p≔\mathrm{expand}\left(\left(y+1\right)x^3+\left(z+4\right)x+3\right)$

$p≔x^{3}y+x^3+xz+4x+3$

$\mathrm{MainVariable}\left(p\,R\right)$

$z$

$\mathrm{Initial}\left(p\,R\right)$

$x$

$\mathrm{MainDegree}\left(p\,R\right)$

$1$

$\mathrm{Rank}\left(p\,R\right)$

$z$

$\mathrm{Tail}\left(p\,R\right)$

$x^{3}y+x^3+4x+3$

$\mathrm{Separant}\left(p\,R\right)$

$x$

$R≔\mathrm{PolynomialRing}\left(\left[z\,y\,x\right]\,3\right)$

$R≔\mathrm{polynomial\_ring}$

$p≔{\left(x+y\right)}^3{z}^3+3z^2+2z+y+4$

$p≔{\left(x+y\right)}^3{z}^3+3z^2+2z+y+4$

$\mathrm{MainVariable}\left(p\,R\right)$

$z$

$\mathrm{Initial}\left(p\,R\right)$

$x^3+y^3$

$\mathrm{MainDegree}\left(p\,R\right)$

$3$

$\mathrm{Rank}\left(p\,R\right)$

$z^3$

$\mathrm{Tail}\left(p\,R\right)$

$y+2z+1$