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DifferentialAlgebra[Tools]

 Initial
 returns the initial of a differential polynomial

 Calling Sequence Initial(ideal, opts) Initial(p, R, opts) Initial(L, R, opts)

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 Initial(ideal), 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 Initial(p,R) returns the initial of p with respect to the ranking of R, or of its embedding polynomial ring, if R is an ideal. It is assumed that p is non-numeric.
 • The function call Initial(L,R) returns the list or the set of the initials of the elements of L with respect to the ranking of R.
 • If ideal is a regular differential chain, the function call Initial(ideal) returns the list of the initials of the chain elements. If ideal is a list of regular differential chains, the function call Initial(ideal) returns a list of lists of initials.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form Initial(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][Initial](...).

Examples

 > with (DifferentialAlgebra): with(Tools):
 > R := DifferentialRing (derivations=[x,y], blocks=[[v,u],p], parameters=[p]);
 ${R}{≔}{\mathrm{differential_ring}}$ (1)
 > Initial (u[x,y]*v[y]-u+p, R);
 ${{v}}_{{y}}$ (2)
 > ideal := RosenfeldGroebner ([u[x]^2-4*u, u[x,y]*v[y]-u+p, v[x,x]-u[x]], R);
 ${\mathrm{ideal}}{≔}\left[{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}\right]$ (3)
 > Equations (ideal[1]);
 $\left[{{v}}_{{x}{,}{x}}{-}{{u}}_{{x}}{,}{p}{}{{u}}_{{x}}{}{{u}}_{{y}}{-}{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{+}{4}{}{u}{}{{v}}_{{y}}{,}{{u}}_{{x}}^{{2}}{-}{4}{}{u}{,}{{u}}_{{y}}^{{2}}{-}{2}{}{u}\right]$ (4)
 > Initial (ideal[1]);
 $\left[{1}{,}{4}{}{u}{,}{1}{,}{1}\right]$ (5)