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DifferentialAlgebra[Tools]

  

LeadingCoefficient

  

returns the leading coefficient of a differential polynomial

 

Calling Sequence

Parameters

Options

Description

Examples

Calling Sequence

LeadingCoefficient(ideal, v, opts)

LeadingCoefficient(p, v, R, opts)

LeadingCoefficient(L, v, R, opts)

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

with(DifferentialAlgebra): with(Tools):

R := DifferentialRing(derivations=[x,y], blocks=[[v,u],p], parameters=[p]);

Rdifferential_ring

(1)

ideal := RosenfeldGroebner([u[x]^2-4*u, u[x,y]*v[y]-u+p, v[x,x]-u[x]], R);

idealregular_differential_chain,regular_differential_chain

(2)

Equations(ideal[1]);

vx,xux,puxuyuuxuy+4uvy,ux24u,uy22u

(3)

The leading coefficients of the chain polynomials, with respect to ux

LeadingCoefficient(ideal[1], u[x]);

−1,puyuuy,1,uy22u

(4)

The derivative is not specified. The initial is returned.

LeadingCoefficient(u[x,y]*v[y]-u+p, R);

vy

(5)

See Also

DifferentialAlgebra

LeadingDerivative

Initial