The function call MainDegree(p,R) returns the main degree of p,   that is, the degree of p with respect to 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 MainDegree(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[MainDegree](..).

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

$x$

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

$y+1$

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

$3$

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

$x^3$

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

$xz+4x+3$

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

$R≔\mathrm{polynomial\_ring}$

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

$p≔x^{3}y+x^3+xz+4x+3$

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

$z$

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

$x$

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

$1$

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

$z$

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

$z$

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

$x^3+y^3$

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

$3$

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

$z^3$

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

$y+2z+1$