returns the tail of a differential polynomial - Maple Help

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DifferentialAlgebra[Tools][Tail] - returns the tail of a differential polynomial

 Calling Sequence Tail(ideal, v, opts) Tail(p, v, R, opts) Tail(L, v, R, opts)

Parameters

 ideal - a differential ideal p - a differential polynomial v (optional) - a variable L - a list or a set of differential polynomials R - a differential polynomial ring or ideal opts (optional) - a sequence of options

Description

 • The function call Tail(p,v,R) returns the tail of p regarded as a univariate polynomial in v, that is the differential polynomial p, regarded as a univariate polynomial in v, minus its leading monomial with respect to this variable, If p does not depend on v then the function call returns $0$.
 • The function call Tail(L,v,R) returns the list or the set of the tails of the elements of L with respect to v.
 • If ideal is a regular differential chain, the function call Tail(ideal,v) returns the list of the tails of the chain elements. If ideal is a list of regular differential chains, the function call Tail(ideal,v) returns a list of lists of tails.
 • When the parameter v is omitted, it is understood to be the leading derivative of the processed differential polynomial with respect to the ranking of R, or the one of its embedding polynomial ring, if R is an ideal. In that case, p must be non-numeric.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form Tail(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][Tail](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$
 > $R:=\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{blocks}=\left[\left[v,u\right],p\right],\mathrm{parameters}=\left[p\right]\right)$
 ${R}{:=}{\mathrm{differential_ring}}$ (1)

The tail, with respect to the leading derivative

 > $\mathrm{Tail}\left({u}_{x,y}{v}_{y}-u+p,R\right)$
 ${-}{u}{+}{p}$ (2)
 > $\mathrm{ideal}:=\mathrm{RosenfeldGroebner}\left(\left[{u}_{x}^{2}-4u,{u}_{x,y}{v}_{y}-u+p,{v}_{x,x}-{u}_{x}\right],R\right)$
 ${\mathrm{ideal}}{:=}\left[{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}\right]$ (3)
 > $\mathrm{Equations}\left({\mathrm{ideal}}_{1}\right)$
 $\left[{{v}}_{{x}{,}{x}}{-}{{u}}_{{x}}{,}{p}{}{{u}}_{{x}}{}{{u}}_{{y}}{-}{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{+}{4}{}{u}{}{{v}}_{{y}}{,}{{u}}_{{x}}^{{2}}{-}{4}{}{u}{,}{{u}}_{{y}}^{{2}}{-}{2}{}{u}\right]$ (4)

The tails of the equations, with respect to ${u}_{x}$

 > $\mathrm{Tail}\left({\mathrm{ideal}}_{1},{u}_{x}\right)$
 $\left[{{v}}_{{x}{,}{x}}{,}{4}{}{{v}}_{{y}}{}{u}{,}{-}{4}{}{u}{,}{0}\right]$ (5)