DifferentialAlgebra[Tools]

 returns the leading coefficient of a differential polynomial

Parameters

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

Options

 • The opts arguments may contain one or more of the options below.
 • fullset = boolean. In the case of the function call LeadingCoefficient(ideal,v), applies the function also over the differential polynomials which state that the derivatives of the parameters are zero. Default value is false.
 • notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of the first argument is used.
 • memout = nonnegative. Specifies a memory limit, in MB, for the computation. Default is zero (no memory out).

Description

 • The function call LeadingCoefficient(p,v,R) returns the leading coefficient of p regarded as a univariate polynomial in v. If p does not depend on v then the function call returns p.
 • The function call LeadingCoefficient(L,v,R) returns the list or the set of the leading coefficients of the elements of L with respect to v.
 • If ideal is a regular differential chain, the function call LeadingCoefficient(ideal,v) returns the list of the leading coefficients of the chain elements. If ideal is a list of regular differential chains, the function call LeadingCoefficient(ideal,v) returns a list of lists of leading coefficients.
 • When the parameter v is omitted, it is understood to be the leading derivative of each processed differential polynomial. In that case, the function behaves as the Initial function.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form LeadingCoefficient(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][LeadingCoefficient](...).

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

The leading coefficients of the chain polynomials, with respect to ${u}_{x}$

 > $\mathrm{LeadingCoefficient}\left(\mathrm{ideal}\left[1\right],u\left[x\right]\right)$
 $\left[{-1}{,}{p}{}{{u}}_{{y}}{-}{u}{}{{u}}_{{y}}{,}{1}{,}{{u}}_{{y}}^{{2}}{-}{2}{}{u}\right]$ (4)

The derivative is not specified. The initial is returned.

 > $\mathrm{LeadingCoefficient}\left(u\left[x,y\right]v\left[y\right]-u+p,R\right)$
 ${{v}}_{{y}}$ (5)