Glossary - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

Differential Algebra Glossary

 

Glossary

Glossary Details

Glossary

• 

This glossary provides definitions for terms that are commonly used in the DifferentialAlgebra package documentation. Certain terms (for example, order and derivative) may have both technical and common meanings. A term is italicized when its technical meaning is used.

– 

attribute: In this package, an attribute is a property of a regular differential chain. Seven attributes are defined: differential, prime, primitive, squarefree, coherent, autoreduced, and normalized. The two first attributes provide properties of the ideal defined by the regular differential chain. The other attributes provide properties of the differential polynomials that constitute the chain.

– 

autoreduced: An attribute of regular differential chains. It indicates that the regular differential chain is autoreduced, in the sense that, for all differential polynomials  and  of the chain, the degree of  in the leading derivative of  is less than the degree of  in its leading derivative.

– 

BLAD (Bibliotheques Lilloises d'Algebre Differentielle): Open source libraries, which are written in the C programming language and dedicated to differential elimination. The DifferentialAlgebra package uses the BLAD libraries for most computations.

– 

block: A list of dependent variables plus a block-keyword. Blocks appear in the definition of rankings.

– 

block-keyword: A keyword, which makes the ranking of a block precise. Five block-keywords are defined: grlexA, grlexB, degrevlexA, degrevlexB, and lex.

– 

characteristic set: A particular case of a regular differential chain.

– 

coherent: An attribute of regular differential chains. It indicates that the regular differential chain is coherent. This concept is described in the Regular Differential Chains section below.

– 

component: In an intersection of differential ideals, presented by regular differential chains, one of the components of the intersection.

– 

constant: An expression whose derivatives are all equal to zero.

– 

degrevlexA: One of the block-keywords.

– 

degrevlexB: One of the block-keywords.

– 

Delta-polynomial: The -polynomial defined by two differential polynomials is important for testing the coherence property of systems of partial differential equations. It plays the same role as the -polynomials of the Groebner bases theory. Let  and  be two differential polynomials, whose leading derivatives  and  are derivatives of some same dependent variable . Denote  and  the derivation operators associated to  and  and  their least common multiple.

• 

If  is different from both  and , then the -polynomial defined by  and  is the differential polynomial    -   , where  and  denote the separants of  and  and the left multiplication by a derivation operator stands for a differentiation.

• 

If  is equal to  (as an example), then the -polynomial defined by  and  is the pseudo-remainder of  by   with respect to .

– 

dependent variable: A variable that depends on the independent variables, for example, a function of the independent variables. In the classical books of differential algebra, it would be called a differential indeterminate.

– 

dependent variable associated to a derivative: If  is a derivative of a dependent variable , then  is said to be the dependent variable associated to .

– 

derivation: In this package, derivations are taken with respect to independent variables and are supposed to commute. The set of the derivations generates the monoid (semigroup) of the derivation operators, which are denoted multiplicatively. In the classical books of differential algebra, derivations are simply abstract operations which satisfy the axioms of derivations.

– 

derivation operator: A power product of independent variables denoting iterated derivations. For example, differentiating a differential polynomial , with respect to the derivation operator , consists of differentiating  twice with respect to , then once with respect to . The identity derivation operator is denoted as .

– 

derivation operator associated to a derivative: If  is a derivative of a dependent variable , then there exists a derivation operator, , such that differentiating  with respect to  gives . The derivation operator  is called the derivation operator associated to .

– 

derivative: A derivative of a dependent variable.

– 

differential: One of the attributes of regular differential chains. It indicates that the ideal defined by the chain is differential. The presence of differential implies the presence of squarefree, and coherent in the partial differential case.

– 

differential algebra: The mathematical theory that provides the theoretical basis for this package.

– 

differential ideal: Mathematically, an ideal which contains the derivatives of some element , whenever it contains . In this package, a differential ideal is a polynomial ideal which is presented either by a single regular differential chain or by a list of regular differential chains. Lists of regular differential chains represent the intersection of the differential ideals defined by the chains. These are called the components of the intersection. The chains in a list must belong to the same differential polynomial ring. The empty list denotes the unit differential ideal. The zero differential ideal can be represented by a regular differential chain. The concept of differential ideals is described in the Differential Ideals section below.

– 

differential indeterminate: An abstract symbol standing for a function, over which the derivations act. In this package, the expression dependent variable is preferred.

– 

differential polynomial: A polynomial, with (usually) rational coefficients, whose variables are either derivatives or independent variables. In some contexts, the field of coefficients may be a non-trivial differential field.

– 

differential ring: Mathematically, a ring endowed with finitely many derivations, which (in our case) are supposed to commute. In this package, a differential ring is a data structure representing a polynomial differential ring, endowed with a ranking, and other minor features. In the case of only one derivation, the ring is said to be ordinary. In the case of two or more derivations, it is said to be partial. In this package, a ring may be endowed with no derivation. In this case, it is no longer differential, and, the use of other packages, such as RegularChains or Groebner is recommended.

– 

grlexA: One of the block-keywords.

– 

grlexB: One of the block-keywords.

– 

inconsistent: An inconsistent system of differential polynomials is a system which has no solution. The differential ideal generated by an inconsistent system is the unit ideal, which is presented, in this package, by the empty list.

– 

independent variable: A variable, with respect to which derivations are taken. Unless stated otherwise, a dependent variable is supposed to depend on all of the independent variables. Common examples are the time , and the space variables , , .

– 

initial: The initial of a non-numeric differential polynomial  is the leading coefficient of , regarded as a univariate polynomial with respect to its leading derivative.

– 

irredundant: A representation of an ideal  as an intersection of ideals , ...,  is said to be irredundant if, given any two different indices  the ideal  is not included in the ideal . In general, the representations computed by the RosenfeldGroebner algorithm are redundant. This issue is addressed in a following section.

– 

leading derivative: The leading derivative  of a non-numeric differential polynomial , is the highest derivative, with respect to some given ranking, such that the degree of  in  is positive. In this package, the leading derivative of differential polynomials which only depend on independent variables is defined.

– 

leading rank: The leading rank of a differential polynomial  is the rank  such that  is the leading derivative of , and,  is the degree of  in . In this package, the leading rank of differential polynomials which do not depend on any derivative is defined. In particular, the leading rank of  is , the one of any other rational number is .

– 

lex: One of the block-keywords.

– 

normalized: An attribute of regular differential chains. It indicates that the regular differential chain is normalized, in the sense that, the initials of the differential polynomials of the chain do not depend on any leading derivative of any element of the chain. The presence of normalized implies the presence of autoreduced and primitive. Regular differential chains which hold these three attributes are canonical representatives of the ideals that they define (in the sense that they only depend on the ideal and on the ranking).

– 

numeric: A numeric differential polynomial is a differential polynomial which does not depend on any derivative or any independent variable.

– 

order: The order of a derivative is the total degree of its associated derivation operator. The order of a differential polynomial is the maximum of the orders of the derivatives it depends on.

– 

orderly: A ranking is orderly if, for all derivatives  and ,  is higher than  whenever the order  of  is greater than the order of .

– 

ordinary: An ordinary differential ring is a ring endowed with a single derivation.

– 

partial: A partial differential ring is a ring endowed with two or more derivations.

– 

prime: An ideal is prime if, whenever it involves some product , it involves at least one the factors. Any prime ideal is radical. A prime differential ideal is a differential ideal, which is prime.

– 

prime: One of the attributes of regular differential chains. It indicates that the ideal defined by the chain is prime. The presence of prime implies the presence of squarefree. The changing of ranking that can be performed using RosenfeldGroebner, for example, only apply to prime ideals. Many computations on regular differential chains are simplified, when the ideal that they define are known to be prime.

– 

primitive: An attribute of regular differential chains. It indicates that each differential polynomial of the chain is primitive, in the sense that the gcd of its coefficients is equal to . Chain differential polynomials are regarded as univariate polynomials in their leading derivatives. Their coefficients are viewed as multivariate polynomials over the ring of the integers.

– 

proper derivative: A derivative  is said to be a proper derivative of a derivative , if  is a derivative of  and is different from .

– 

radical: An ideal  of a ring  is said to be radical if it involves some element  whenever it involves any power  of , where  is any non-negative integer. The radical of an ideal  is the set of all the elements  of , a power of which belongs to . The radical of an ideal is an ideal. The radical of a differential ideal is a differential ideal.

– 

rank: A derivative, or, an independent variable, raised to some positive integer. In addition, the two special ranks  and  are defined. Any ranking extends to a total ordering on ranks:  is less than , which is less than any other rank. Two ranks  and  are compared by comparing, first,  and , then, the exponents  and .

– 

ranking: Any total ordering over the set of the derivatives, which satisfies the two axioms of rankings. In this package, rankings are extended to the set of the independent variables. This concept is described in the Rankings section below.

– 

redundant: Not irredundant. See the Irredundant definition above.

– 

redundant component: In an intersection of a differential ideal, a component which contains another component.

– 

regular differential chain: A data structure containing a list of differential polynomials sorted by increasing rank, plus some minor features. A few variants of regular differential chain are implemented. These variants can be selected by customizing the attributes of the chain. This concept is described in the Regular Differential Chains section below.

– 

saturation: If  is an ideal of a ring  and  is a multiplicative family of , then the saturation of  by  is the set  of all the elements  of  such that,  belongs to , for some  of . The saturation of an ideal is an ideal. The saturation of a differential ideal is a differential ideal.

– 

separant: The separant of a non-numeric differential polynomial , is the partial derivative of  with respect to its leading derivative.

– 

squarefree: An attribute of regular differential chains. It indicates that the regular differential chain is squarefree. This concept is described in the Regular Differential Chains section.

– 

tail: The tail of a non-numeric differential polynomial , is equal to  where  is the initial of  and  is its leading rank.

Glossary Details

  

This section describes some of the concepts mentioned in the glossary in more detail.

Rankings

• 

A ranking is any total ordering over the set of the derivatives (in this package, rankings are extended to the independent variables), which satisfies the two axioms of rankings:

– 

Each derivative  is less than any of its proper derivatives.

– 

If  and  are derivatives, such that  is less than  and  is any independent variable, then the derivative of  with respect to  is less than the derivative of  with respect to .

• 

In this package, rankings are defined by the list  > ... >  of the independent variables plus a list  >> ... >>  of blocks. Each block  is defined by a list  > ... >  of dependent variables plus a block-keyword, which is grlexA, grlexB, degrevlexA, degrevlexB or lex. Any dependent variable must appear in exactly one block.

• 

The >> operator between blocks indicates a block elimination ranking: if  >>  are two blocks,  is any derivative of any dependent variable occurring in , and,  is any derivative of any dependent variable occurring in , then  > .

• 

Within a given block  =  > ... > , in the ordinary differential case (only one derivation), the derivatives are ordered by the unique orderly ranking such that  > ... > .

• 

Within a given block  =  > ... > , in the partial differential case (two or more derivations), the block-keyword of  is necessary to precise the ranking. Consider two derivatives  and  of two dependent variables  and , such that  >= . Denote  and  as the derivation operators associated to these two derivatives. In the sequel,  (lex) means that  with respect to the lexicographical order of the Groebner bases theory, defined by  > ... > , while,  (degrevlex) means that  with respect to the degree reverse lexicographic order. Depending on the block-keyword of , listed below, one has  if the following condition is satisfied:

– 

grlexA.  If the order of  is greater than the one of  else, if the orders are equal and  else, if the orders are equal and  and  (lex).

– 

grlexB. If the order of  is greater than the one of  else, if the orders are equal and  (lex) else, if  and .

– 

degrevlexA. If the order of  is greater than the one of  else, if the orders are equal and  else, if the orders are equal and  and  (degrevlex).

– 

degrevlexB. If the order of  is greater than the one of  else, if the orders are equal and  (degrevlex) else, if  and .

– 

lex. If  (lex) else, if  and .

Regular Differential Chains

• 

Regular differential chains generalize the regular chains of the non-differential polynomial algebras (see the RegularChains package and [ALM99]) and the characteristic sets of classical differential algebra. In the sequel, one assumes that a ranking is fixed, and therefore that each non-numeric differential polynomial admits a leading derivative.

• 

A regular differential chain appears as a list of differential polynomials , ...,  with rational numbers for coefficients. Each differential polynomial  admits a leading derivative,  which is a derivative. The list , ...,  is sorted by increasing leading rank.

• 

The set , ...,  is differentially triangular and partially autoreduced:

– 

Given any two different indices , the leading derivative  is not a derivative of .

– 

Given any two indices , the differential polynomial  does not depend on any proper derivative of .

• 

At this stage, one needs to introduce the polynomial ring  =  [, ..., ], which is obtained by moving into the base field, all the independent variables and derivatives, the  depend on, but which are not the leading derivative of any .

• 

The set , ...,  is a squarefree regular chain:

– 

Given any index , the initial of  is an invertible element of / where  denotes the ideal generated by , ...,  in the ring .

– 

Given any index , the separant of  is an invertible element of / where  denotes the ideal generated by , ...,  in the ring .

• 

At this stage, the polynomial system  must be introduced, where  is a derivative. It is the set of all the derivatives of the differential polynomials  whose leading derivatives are strictly less than . The ring  must also be introduced. It is obtained by moving into the base field of the polynomials, all the independent variables and the derivatives, the elements of  depend on, but which are not the leading derivative of any element of .

• 

In the case of partial differential systems, the set , ...,  is coherent: given any two different indices , such that  and  are derivatives of some same dependent variable , the -polynomial defined by  and  is zero in the ring / where  is least common derivative of  and  and  denotes the ideal generated by  in .

Differential Ideals

• 

Every regular differential chain , ..., , of a polynomial differential ring , defines a differential ideal , which is the set of all the differential polynomials  such that, for some power product  of the initials and the separants of the , the differential polynomial  is a finite linear combination of the derivatives of the , with differential polynomials of  for coefficients.

• 

One stresses the fact the chain , ...,  is not a generating family of . However, it completely defines  and permits to compute a normal form of the residue class of any differential polynomial in /. See NormalForm. In particular:

– 

It permits to decide membership in , that is, zero in /: a differential polynomial  belongs to  if and only if its normal form is .

– 

It permits to decide if a differential polynomial  is regular modulo , that is, a non-zero divisor in /.

• 

Given any system of differential polynomials , ...,  and any ranking, it is possible to compute a representation of the radical  of the differential ideal generated by this system, as a finite intersection of radical differential ideals , ..., , presented by regular differential chains , ..., . See RosenfeldGroebner [BLOP95,BLOP09]. See also [W98,LW99,H00,BKM01]. This representation permits to decide membership in : a differential polynomial  belongs to  if and only if it belongs to each differential ideal , i.e., if and only if its normal forms, with respect to all the regular differential chains , are all . See BelongsTo.

• 

One stresses the fact that, in general, the computed representation , ...,  is redundant. Indeed, the problem of deciding whether two differential ideals presented by regular differential chains are included in each other, is still open. In the particular case of a differential ideal generated by a single differential polynomial, this problem is, however, solved, thanks to the Low Power Theorem. See RosenfeldGroebner and its singsol = essential option [H99].

• 

During the decomposition process, in general, the coefficients of the differential polynomials are assumed to lie in the field  (, ..., ) obtained by adjoining the independent variables to the field of the rational numbers. In particular, the computed regular differential chains do not involve any differential polynomial which only depends on the independent variables. In this package, it is, however, possible to compute decompositions of radical differential ideals generated by differential polynomials with coefficients in more sophisticated differential fields.

Normal Forms

• 

Let  be a regular differential chain of a differential polynomial ring  and  be a differential polynomial of . Let  be the differential ideal defined by . The normal form of  with respect to  is a rational differential fraction / such that

– 

 is regular (that is, a non-zero divisor) in /.

– 

/ is equivalent to  modulo , in the sense that  belongs to .

– 

 is reduced with respect to , in the sense that, given any leading rank  of any element of , it does not depend on any proper derivative of , and, has degree less than , in .

– 

 does not depend on any derivative of , where  denotes any leading derivative of any element of .

• 

The normal form / of  with respect to  is a canonical representative of the residue class of  in /, in the sense that

– 

/ is  if and only if  belongs to .

– 

If  is equivalent to  modulo , then the normal forms of  and , with respect to , are equal.

• 

The normal form of  can be computed by means of the NormalForm function [BL00].

• 

Let us extend the above definition and consider a rational differential fraction /, where  and  are differential polynomials of . The NormalForm function can be used to compute a normal form of /, with respect to . If it succeeds, then it returns a rational differential fraction / such that

– 

 is regular (that is, a non-zero divisor) in /.

– 

/ is equivalent to / modulo , in the sense that  belongs to .

– 

 is reduced with respect to .

– 

 does not depend on any derivative of any leading derivative of .

• 

If it succeeds, then  is regular in /, and the normal form / is a canonical representative of the residue class of / in the total fraction ring of /, in the sense that

– 

/ is  if and only if  belongs to .

– 

If / is equivalent to / modulo  (with  regular modulo ), then the normal forms of / and / with respect to  are equal.

• 

The NormalForm function may fail to compute a normal form of / in the following cases:

– 

If  is zero in /.

– 

If  is a zero-divisor in /.

– 

If the function is led to invert another zero-divisor in /, during the normal form computation.

• 

However, for each rational differential fraction /, it is possible to split  into finitely many regular differential chains , ..., , , ...,  such that, denoting  the radical differential ideal defined by

– 

The normal form of / can be computed, with respect to , for  <= k <= .

– 

 is zero in /, for  <=  <= .

– 

The differential ideal  is equal to the intersection of the differential ideals , for  <= k <= ,

• 

This decomposition can be achieved by using the NormalForm function.

See Also

DifferentialAlgebra

 


Download Help Document