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DifferentialAlgebra[Tools][PreparationEquation] - returns the preparation equation of a differential polynomial

Calling Sequence

PreparationEquation (f, regchain, opts)

Parameters

f

-

a differential polynomial

regchain

-

a regular differential chain

opts (optional)

-

a sequence of options

Description

• 

The function call PreparationEquation (f, regchain) returns a preparation equation [K73, chapter IV, section 13] for f with respect to regchain. The argument f is regarded as a differential polynomial of the embedding ring of regchain.

• 

Let I denote the differential ideal defined by regchain and denote A1, ..., Ar the differential polynomials which constitute the chain. Introduce r new dependent variables zi. Each variable zi represents the differential polynomial Ai.

• 

The returned preparation equation is an expression having the form hf = c1t1 + ... + cntn. The differential polynomial h is a power product of initials and separants of the Ak. The coefficients ci are reduced and regular with respect to I. The monomials ti are power products of the zk variables and their derivatives. They satisfy some further properties, described in [K73, chapter IV, section 13]. If each zk is replaced by the corresponding Ak, in all terms ti, then the preparation equation becomes a true equality.

• 

Preparation equations are an important tool in the context of the Low Power Theorem. See RosenfeldGroebner with the option singsol = essential.

• 

This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form PreparationEquation(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][PreparationEquation](...).

Examples

withDifferentialAlgebra:withTools:

Basic illustration

• 

The following examples illustrate the function, syntactically.

R:=DifferentialRingderivations=t,blocks=u,s,c

R:=differential_ring

(1)

ideal:=PretendRegularDifferentialChainc2+s21,ct+s,u,R,pretend=false

ideal:=regular_differential_chain

(2)

Equationsideal

u,ct+s,c2+s21

(3)

f:=ct+sut+ut2

f:=ct+sut+ut2

(4)

prepeq:=PreparationEquationf,ideal

prepeq:=sut+ctut+ut2=z2z3t+z3t2

(5)
• 

If one substitutes the Ai to the zi, the equation becomes an equality.

expandevalprepeq,z3=u,z2=ct+s

sut+ctut+ut2=sut+ctut+ut2

(6)
• 

Changing the identifier for the zi.

PreparationEquationf,ideal,zstring=A(%d)

sut+ctut+ut2=A3t2+A2A3t

(7)
• 

Since all monomials ti have degree q=2, the preparation congruence is equal to the preparation equation.

PreparationEquationf,ideal,congruence=true

sut+ctut+ut2=z2z3t+z3t2

(8)
• 

However, if the two first elements of the regular differential chain are considered as base field defining equations, then, only one monomial ti is left in the congruence.

PreparationEquationf,ideal,congruence=true,n=2

sut+ctut+ut2=z3t2

(9)

The Low Power Theorem

• 

The next example illustrates the Low Power Theorem. See [R50, chapter III] and [K73, chapter IV, section 15].

R:=DifferentialRingderivations=t,blocks=y

R:=differential_ring

(10)

f:=yt34ytyt+8y2

f:=4tyyt+yt3+8y2

(11)
• 

First compute a representation of the radical of the differential ideal generated by f, by means of RosenfeldGroebner.

ideal:=RosenfeldGroebnerf,R

ideal:=regular_differential_chain,regular_differential_chain,regular_differential_chain

(12)

Equationsideal

4tyyt+yt3+8y2,4t3+27y,y

(13)
• 

Second, remove any regular differential chain which involve two or more differential polynomials, by application of the Component Theorem [K73, chapter IV, section 14]. In our case, no regular differential chain is removed by this process. Third, compute a preparation congruence for f, with respect to each of the two singular components, i.e., the two last ones.

• 

In the first case, there is only one monomial t1, of the form z1q. Thus this regular differential chain must be kept in the decomposition.

rhsPreparationEquationf,ideal2,congruence=true

167365651248t3z1

(14)
• 

In the second case, the right hand-side of the preparation congruence involves two monomials. Thus this regular differential chain is redundant.

rhsPreparationEquationf,ideal3,congruence=true

4tz1z1t+8z12

(15)
• 

Indeed, RosenfeldGroebner with the option singsol = essential removes the second singular component from the decomposition.

ideal:=RosenfeldGroebnerf,R,singsol=essential

ideal:=regular_differential_chain,regular_differential_chain

(16)

Equationsideal

4tyyt+yt3+8y2,4t3+27y

(17)

See Also

DifferentialAlgebra, RosenfeldGroebner


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