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RegularChains

 Initial
 initial of a nonconstant polynomial

 Calling Sequence Initial(p, R)

Parameters

 R - polynomial ring p - polynomial of R

Description

 • The function call Initial(p,R) returns the initial of p with respect to the variable ordering of R, that is, the leading coefficient 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 Initial(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[Initial](..).

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)
 > $\mathrm{Separant}\left(p,R\right)$
 ${3}{}{{x}}^{{2}}{}{y}{+}{3}{}{{x}}^{{2}}{+}{z}{+}{4}$ (8)

Change the ordering of the variable.

 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (9)
 > $p≔\mathrm{expand}\left(\left(y+1\right){x}^{3}+\left(z+4\right)x+3\right)$
 ${p}{≔}{{x}}^{{3}}{}{y}{+}{{x}}^{{3}}{+}{x}{}{z}{+}{4}{}{x}{+}{3}$ (10)
 > $\mathrm{MainVariable}\left(p,R\right)$
 ${z}$ (11)
 > $\mathrm{Initial}\left(p,R\right)$
 ${x}$ (12)
 > $\mathrm{MainDegree}\left(p,R\right)$
 ${1}$ (13)
 > $\mathrm{Rank}\left(p,R\right)$
 ${z}$ (14)
 > $\mathrm{Tail}\left(p,R\right)$
 ${{x}}^{{3}}{}{y}{+}{{x}}^{{3}}{+}{4}{}{x}{+}{3}$ (15)
 > $\mathrm{Separant}\left(p,R\right)$
 ${x}$ (16)

Set the characteristic to 3.

 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right],3\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (17)
 > $p≔{\left(x+y\right)}^{3}{z}^{3}+3{z}^{2}+2z+y+4$
 ${p}{≔}{\left({x}{+}{y}\right)}^{{3}}{}{{z}}^{{3}}{+}{3}{}{{z}}^{{2}}{+}{2}{}{z}{+}{y}{+}{4}$ (18)
 > $\mathrm{MainVariable}\left(p,R\right)$
 ${z}$ (19)
 > $\mathrm{Initial}\left(p,R\right)$
 ${{x}}^{{3}}{+}{{y}}^{{3}}$ (20)
 > $\mathrm{MainDegree}\left(p,R\right)$
 ${3}$ (21)
 > $\mathrm{Rank}\left(p,R\right)$
 ${{z}}^{{3}}$ (22)
 > $\mathrm{Tail}\left(p,R\right)$
 ${y}{+}{2}{}{z}{+}{1}$ (23)