The function call Separant(p,R) returns the separant of p with respect to the variable ordering of R, that is, the derivative of p with respect to its main variable.

It is assumed that p is nonconstant polynomial, and that the characteristic of R is zero.

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

$x$

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

$p≔x^{3}y+x^3+xz+4x+3$

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

$z$

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

$x$