The function call MainVariable(p,R) returns the greatest variable of p with respect to the variable ordering of R.

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 MainVariable(..) only after executing the command with(RegularChains).  However, it can always be accessed through the long form of the command by using RegularChains[MainVariable](..).

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

$x$

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

$y+1$

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

$3$

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

$x^3$

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

$xz+4x+3$

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

$R≔\mathrm{polynomial\_ring}$

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

$p≔x^{3}y+x^3+xz+4x+3$

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

$z$

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

$x$

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

$1$

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

$z$

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

$z$

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

$x^3+y^3$

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

$3$

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

$z^3$

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

$y+2z+1$