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RegularChains

 Rank
 rank of a nonconstant polynomial

 Calling Sequence Rank(p, R)

Parameters

 R - polynomial ring p - polynomial of R

Description

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

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $p≔\left(y+1\right){x}^{3}+\left(z+4\right)x+3$
 ${p}{≔}\left({y}{+}{1}\right){}{{x}}^{{3}}{+}\left({z}{+}{4}\right){}{x}{+}{3}$ (2)
 > $\mathrm{MainVariable}\left(p,R\right)$
 ${x}$ (3)
 > $\mathrm{Initial}\left(p,R\right)$
 ${y}{+}{1}$ (4)
 > $\mathrm{MainDegree}\left(p,R\right)$
 ${3}$ (5)
 > $\mathrm{Rank}\left(p,R\right)$
 ${{x}}^{{3}}$ (6)
 > $\mathrm{Tail}\left(p,R\right)$
 ${x}{}{z}{+}{4}{}{x}{+}{3}$ (7)

Change the ordering of the variable.

 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (8)
 > $p≔\mathrm{expand}\left(\left(y+1\right){x}^{3}+\left(z+4\right)x+3\right)$
 ${p}{≔}{{x}}^{{3}}{}{y}{+}{{x}}^{{3}}{+}{x}{}{z}{+}{4}{}{x}{+}{3}$ (9)
 > $\mathrm{MainVariable}\left(p,R\right)$
 ${z}$ (10)
 > $\mathrm{Initial}\left(p,R\right)$
 ${x}$ (11)
 > $\mathrm{MainDegree}\left(p,R\right)$
 ${1}$ (12)
 > $\mathrm{Rank}\left(p,R\right)$
 ${z}$ (13)
 > $\mathrm{Tail}\left(p,R\right)$
 ${{x}}^{{3}}{}{y}{+}{{x}}^{{3}}{+}{4}{}{x}{+}{3}$ (14)

Set the characteristic to 3.

 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right],3\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (15)
 > $p≔{\left(x+y\right)}^{3}{z}^{3}+3{z}^{2}+2z+y+4$
 ${p}{≔}{\left({x}{+}{y}\right)}^{{3}}{}{{z}}^{{3}}{+}{3}{}{{z}}^{{2}}{+}{2}{}{z}{+}{y}{+}{4}$ (16)
 > $\mathrm{MainVariable}\left(p,R\right)$
 ${z}$ (17)
 > $\mathrm{Initial}\left(p,R\right)$
 ${{x}}^{{3}}{+}{{y}}^{{3}}$ (18)
 > $\mathrm{MainDegree}\left(p,R\right)$
 ${3}$ (19)
 > $\mathrm{Rank}\left(p,R\right)$
 ${{z}}^{{3}}$ (20)
 > $\mathrm{Tail}\left(p,R\right)$
 ${y}{+}{2}{}{z}{+}{1}$ (21)

 See Also