DifferentialAlgebra[Tools] - Maple Programming Help

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DifferentialAlgebra[Tools]

 returns the leading rank of a differential polynomial

Parameters

 ideal - a differential ideal p - a differential polynomial 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 LeadingRank(ideal), applies the function also over the differential polynomials which state that the derivatives of the parameters are zero. Default value is false. This option is incompatible with the diff notation.
 • 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 LeadingRank(p,R) returns the leading rank of p with respect to the ranking of R, or of its embedding ring, if R is an ideal.
 • The function is extended to numeric polynomials: the leading rank of $0$ is $0$. The leading rank of any nonzero numerical polynomial is $1$. It is also extended to differential polynomials which involve independent variables only.
 • The function call LeadingRank(L,R) returns the list or the set of the leading ranks of the elements of L with respect to the ranking of R.
 • If ideal is a regular differential chain, the function call LeadingRank(ideal) returns the list of the leading ranks of the chain elements. If ideal is a list of regular differential chains, the function call LeadingRank(ideal) returns a list of lists of leading ranks.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form LeadingRank(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][LeadingRank](...).

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)
 > $\mathrm{LeadingRank}\left({u}_{x,y}{v}_{y}-u+p,R\right)$
 ${{u}}_{{x}{,}{y}}$ (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)
 > $\mathrm{LeadingRank}\left({\mathrm{ideal}}_{1}\right)$
 $\left[{{v}}_{{x}{,}{x}}{,}{{v}}_{{y}}{,}{{u}}_{{x}}^{{2}}{,}{{u}}_{{y}}^{{2}}\right]$ (5)
 > $\mathrm{LeadingRank}\left(\left[0,421\right],R\right)$
 $\left[{0}{,}{1}\right]$ (6)