Differential Algebra Glossary

Glossary


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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.

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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.

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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.

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

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

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

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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.

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

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

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degrevlexA: One of the blockkeywords.

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degrevlexB: One of the blockkeywords.

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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.

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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.

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derivative: A derivative of a dependent variable.

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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.

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

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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.

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

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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 nontrivial differential field.

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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.

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grlexA: One of the blockkeywords.

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grlexB: One of the blockkeywords.

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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.

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

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lex: One of the blockkeywords.

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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).

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

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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.

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

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

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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.

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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.

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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.

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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.

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redundant: Not irredundant. See the Irredundant definition above.

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

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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.

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

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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.



Glossary Details



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


Rankings


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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:

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Each derivative is less than any of its proper derivatives.



Regular Differential Chains


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Regular differential chains generalize the regular chains of the nondifferential 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 nonnumeric differential polynomial admits a leading derivative.

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

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



Differential Ideals


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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.

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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].



Normal Forms


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

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is reduced with respect to .

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

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

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

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This decomposition can be achieved by using the NormalForm function.



