Differential Algebra Glossary
GlossaryGlossary Details
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk0">Glossary</Text-field>
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 NiNJInBHNiI= and NiNJInFHNiI= of the chain, the degree of NiNJInBHNiI= in the leading derivative of NiNJInFHNiI= is less than the degree of NiNJInFHNiI= 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 NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmRGVsdGE7Rik=-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 NiNJIlNHNiI=-polynomials of the Groebner bases theory. Let NiMmSSJBRzYiNiNJImlHRiU= and NiMmSSJBRzYiNiNJImpHRiU= be two differential polynomials, whose leading derivatives NiMmSSJ2RzYiNiNJImlHRiU= and NiMmSSJ2RzYiNiNJImpHRiU= are derivatives of some same dependent variable NiNJInVHNiI=. Denote NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImlGKQ== and NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImpGKQ== the derivation operators associated to NiMmSSJ2RzYiNiNJImlHRiU= and NiMmSSJ2RzYiNiNJImpHRiU= and NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUklbXJvd0c2JEYmL0YoRio2JS1GLzYjUSJpRiktSSNtb0c2JEYmL0YoRio2I1EoJmNvbW1hO0YpLUYvNiNRImpGKQ== their least common multiple.
If NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUklbXJvd0c2JEYmL0YoRio2JS1GLzYjUSJpRiktSSNtb0c2JEYmL0YoRio2I1EoJmNvbW1hO0YpLUYvNiNRImpGKQ== is different from both NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImlGKQ== and NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImpGKQ==, then the NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmRGVsdGE7Rik=-polynomial defined by NiMmSSJBRzYiNiNJImlHRiU= and NiMmSSJBRzYiNiNJImpHRiU= is the differential polynomial NiMmSSJTRzYiNiNJImlHRiU= NiMtSSZtZnJhY0c2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYkLUklbXN1Ykc2JEYmL0YoRio2JC1JI21pRzYkRiYvRihGKjYjUSgmdGhldGE7RiktSSVtcm93RzYkRiYvRihGKjYlLUY0NiNRImlGKS1JI21vRzYkRiYvRihGKjYjUSgmY29tbWE7RiktRjQ2I1EiakYpLUYvNiRGM0ZH NiMmSSJBRzYiNiNJImpHRiU= - NiMmSSJTRzYiNiNJImpHRiU= NiMtSSZtZnJhY0c2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYkLUklbXN1Ykc2JEYmL0YoRio2JC1JI21pRzYkRiYvRihGKjYjUSgmdGhldGE7RiktSSVtcm93RzYkRiYvRihGKjYlLUY0NiNRImlGKS1JI21vRzYkRiYvRihGKjYjUSgmY29tbWE7RiktRjQ2I1EiakYpLUYvNiRGM0Y+ NiMmSSJBRzYiNiNJImlHRiU=, where NiMmSSJTRzYiNiNJImlHRiU= and NiMmSSJTRzYiNiNJImpHRiU= denote the separants of NiMmSSJBRzYiNiNJImlHRiU= and NiMmSSJBRzYiNiNJImpHRiU= and the left multiplication by a derivation operator stands for a differentiation.
If NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUklbXJvd0c2JEYmL0YoRio2JS1GLzYjUSJpRiktSSNtb0c2JEYmL0YoRio2I1EoJmNvbW1hO0YpLUYvNiNRImpGKQ== is equal to NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImpGKQ== (as an example), then the NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmRGVsdGE7Rik=-polynomial defined by NiMmSSJBRzYiNiNJImlHRiU= and NiMmSSJBRzYiNiNJImpHRiU= is the pseudo-remainder of NiMmSSJBRzYiNiNJImpHRiU= by NiMtSSZtZnJhY0c2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYkLUklbXN1Ykc2JEYmL0YoRio2JC1JI21pRzYkRiYvRihGKjYjUSgmdGhldGE7RiktRjQ2I1EiakYpLUYvNiRGMy1GNDYjUSJpRik= NiMmSSJBRzYiNiNJImlHRiU= with respect to NiMmSSJ2RzYiNiNJImpHRiU=.
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 NiNJInZHNiI= is a derivative of a dependent variable NiNJInVHNiI=, then NiNJInVHNiI= is said to be the dependent variable associated to NiNJInZHNiI=.
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 NiNJInBHNiI=, with respect to the derivation operator NiMqJkkieEc2IiIiI0kieUdGJSIiIg==, consists of differentiating NiNJInBHNiI= twice with respect to NiNJInhHNiI=, then once with respect to NiNJInlHNiI=. The identity derivation operator is denoted as NiMiIiI=.
derivation operator associated to a derivative: If NiNJInZHNiI= is a derivative of a dependent variable NiNJInVHNiI=, then there exists a derivation operator, NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmdGhldGE7Rik=, such that differentiating NiNJInVHNiI= with respect to NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmdGhldGE7Rik= gives NiNJInZHNiI=. The derivation operator NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmdGhldGE7Rik= is called the derivation operator associated to NiNJInZHNiI=.
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 NiNJInBHNiI=, whenever it contains NiNJInBHNiI=. 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 NiNJInRHNiI=, and the space variables NiNJInhHNiI=, NiNJInlHNiI=, NiNJInpHNiI=.
initial: The initial of a non-numeric differential polynomial NiNJInBHNiI= is the leading coefficient of NiNJInBHNiI=, regarded as a univariate polynomial with respect to its leading derivative.
irredundant: A representation of an ideal NiNJIkpHNiI= as an intersection of ideals NiMmXiMiIiI2I0Yl, ..., NiMmXiMiIiI2I0kibkc2Ig== is said to be irredundant if, given any two different indices NiRJImpHNiJJImtHRiQ= the ideal NiMmXiMiIiI2I0kiakc2Ig== is not included in the ideal NiMmXiMiIiI2I0kia0c2Ig==. In general, the representations computed by the RosenfeldGroebner algorithm are redundant. This issue is addressed in a following section.
leading derivative: The leading derivative NiNJInZHNiI= of a non-numeric differential polynomial NiNJInBHNiI=, is the highest derivative, with respect to some given ranking, such that the degree of NiNJInBHNiI= in NiNJInZHNiI= 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 NiNJInBHNiI= is the rank NiMpSSJ2RzYiSSJkR0Yl such that NiNJInZHNiI= is the leading derivative of NiNJInBHNiI=, and, NiNJImRHNiI= is the degree of NiNJInBHNiI= in NiNJInZHNiI=. In this package, the leading rank of differential polynomials which do not depend on any derivative is defined. In particular, the leading rank of NiMiIiE= is NiMiIiE=, the one of any other rational number is NiMiIiI=.
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 NiNJInVHNiI= and NiNJInZHNiI=, NiNJInVHNiI= is higher than NiNJInZHNiI= whenever the order of NiNJInVHNiI= is greater than the order of NiNJInZHNiI=.
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 NiMqJkkicEc2IiIiIkkicUdGJUYm, 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 NiMiIiI=. 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 NiNJInZHNiI= is said to be a proper derivative of a derivative NiNJInVHNiI=, if NiNJInZHNiI= is a derivative of NiNJInVHNiI= and is different from NiNJInVHNiI=.
radical: An ideal NiNeIyIiIg== of a ring NiNJIlJHNiI= is said to be radical if it involves some element NiNJInBHNiI= whenever it involves any power NiMpSSJwRzYiSSJkR0Yl of NiNJInBHNiI=, where NiNJImRHNiI= is any non-negative integer. The radical of an ideal NiNeIyIiIg== is the set of all the elements NiNJInBHNiI= of NiNJIlJHNiI=, a power of which belongs to NiNeIyIiIg==. 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 NiMiIiE= and NiMiIiI= are defined. Any ranking extends to a total ordering on ranks: NiMiIiE= is less than NiMiIiI=, which is less than any other rank. Two ranks NiMpSSJ1RzYiSSJuR0Yl and NiMpSSJ2RzYiSSJtR0Yl are compared by comparing, first, NiNJInVHNiI= and NiNJInZHNiI=, then, the exponents NiNJIm5HNiI= and NiNJIm1HNiI=.
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 NiNeIyIiIg== is an ideal of a ring NiNJIlJHNiI= and NiNJIk1HNiI= is a multiplicative family of NiNJIlJHNiI=, then the saturation of NiNeIyIiIg== by NiNJIk1HNiI= is the set NiNJIkpHNiI= of all the elements NiNJInBHNiI= of NiNJIlJHNiI= such that, NiMqJkkibUc2IiIiIkkicEdGJUYm belongs to NiNeIyIiIg==, for some NiNJIm1HNiI= of NiNJIk1HNiI=. 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 NiNJInBHNiI=, is the partial derivative of NiNJInBHNiI= 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 NiNJInBHNiI=, is equal to NiMsJkkicEc2IiIiIiomSSJjR0YlRiYpSSJ2R0YlSSJkR0YlRiYhIiI= where NiNJImNHNiI= is the initial of NiNJInBHNiI= and NiMpSSJ2RzYiSSJkR0Yl is its leading rank.
<Text-field style="Heading 2" layout="Heading 2" bookmark="bkmrk1">Glossary Details </Text-field>
This section describes some of the concepts mentioned in the glossary in more detail.
<Text-field style="Heading 3" layout="Heading 3" bookmark="Rankings">Rankings</Text-field>
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 NiNJInVHNiI= is less than any of its proper derivatives.
If NiNJInVHNiI= and NiNJInZHNiI= are derivatives, such that NiNJInVHNiI= is less than NiNJInZHNiI= and NiNJInhHNiI= is any independent variable, then the derivative of NiNJInVHNiI= with respect to NiNJInhHNiI= is less than the derivative of NiNJInZHNiI= with respect to NiNJInhHNiI=.
In this package, rankings are defined by the list NiMmSSJ4RzYiNiMiIiI= > ... > NiMmSSJ4RzYiNiNJInBHRiU= of the independent variables plus a list NiMmSSJiRzYiNiMiIiI= >> ... >> NiMmSSJiRzYiNiNJIm5HRiU= of blocks. Each block NiNJImJHNiI= is defined by a list NiMmSSJ1RzYiNiMiIiI= > ... > NiMmSSJ1RzYiNiNJIm1HRiU= 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 NiMmSSJiRzYiNiNJImlHRiU= >> NiMmSSJiRzYiNiNJImpHRiU= are two blocks, NiMmSSJ2RzYiNiNJImlHRiU= is any derivative of any dependent variable occurring in NiMmSSJiRzYiNiNJImlHRiU=, and, NiMmSSJ2RzYiNiNJImpHRiU= is any derivative of any dependent variable occurring in NiMmSSJiRzYiNiNJImpHRiU=, then NiMmSSJ2RzYiNiNJImlHRiU= > NiMmSSJ2RzYiNiNJImpHRiU=.
Within a given block NiNJImJHNiI= = NiMmSSJ1RzYiNiMiIiI= > ... > NiMmSSJ1RzYiNiNJIm1HRiU=, in the ordinary differential case (only one derivation), the derivatives are ordered by the unique orderly ranking such that NiMmSSJ1RzYiNiMiIiI= > ... > NiMmSSJ1RzYiNiNJIm1HRiU=.
Within a given block NiNJImJHNiI= = NiMmSSJ1RzYiNiMiIiI= > ... > NiMmSSJ1RzYiNiNJIm1HRiU=, in the partial differential case (two or more derivations), the block-keyword of NiNJImJHNiI= is necessary to precise the ranking. Consider two derivatives NiMmSSJ2RzYiNiNJImlHRiU= and NiMmSSJ2RzYiNiNJImpHRiU= of two dependent variables NiMmSSJ1RzYiNiNJImlHRiU= and NiMmSSJ1RzYiNiNJImpHRiU=, such that NiNJImlHNiI= >= NiNJImpHNiI=. Denote NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImlGKQ== and NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EoJnRoZXRhO0YpLUYvNiNRImpGKQ== as the derivation operators associated to these two derivatives. In the sequel, NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (lex) means that NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= with respect to the lexicographical order of the Groebner bases theory, defined by NiMmSSJ4RzYiNiMiIiI= > ... > NiMmSSJ4RzYiNiNJInBHRiU=, while, NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (degrevlex) means that NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= with respect to the degree reverse lexicographic order. Depending on the block-keyword of NiNJImJHNiI=, listed below, one has NiMyJkkidkc2IjYjSSJqR0YmJkYlNiNJImlHRiY= if the following condition is satisfied:
grlexA. If the order of NiMmSSJ2RzYiNiNJImlHRiU= is greater than the one of NiMmSSJ2RzYiNiNJImpHRiU= else, if the orders are equal and NiMySSJpRzYiSSJqR0Yl else, if the orders are equal and NiMvSSJpRzYiSSJqR0Yl and NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (lex).
grlexB. If the order of NiMmSSJ2RzYiNiNJImlHRiU= is greater than the one of NiMmSSJ2RzYiNiNJImpHRiU= else, if the orders are equal and NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (lex) else, if NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJpRiktSSNtb0c2JEYmL0YoRio2I1EpJmVxdWFscztGKS1GLzYkRjMtRjQ2I1EiakYp and NiMySSJpRzYiSSJqR0Yl.
degrevlexA. If the order of NiMmSSJ2RzYiNiNJImlHRiU= is greater than the one of NiMmSSJ2RzYiNiNJImpHRiU= else, if the orders are equal and NiMySSJpRzYiSSJqR0Yl else, if the orders are equal and NiMvSSJpRzYiSSJqR0Yl and NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (degrevlex).
degrevlexB. If the order of NiMmSSJ2RzYiNiNJImlHRiU= is greater than the one of NiMmSSJ2RzYiNiNJImpHRiU= else, if the orders are equal and NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (degrevlex) else, if NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJpRiktSSNtb0c2JEYmL0YoRio2I1EpJmVxdWFscztGKS1GLzYkRjMtRjQ2I1EiakYp and NiMySSJpRzYiSSJqR0Yl.
lex. If NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJqRiktSSNtb0c2JEYmL0YoRio2I1ElJmx0O0YpLUYvNiRGMy1GNDYjUSJpRik= (lex) else, if NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GNDYjUSJpRiktSSNtb0c2JEYmL0YoRio2I1EpJmVxdWFscztGKS1GLzYkRjMtRjQ2I1EiakYp and NiMySSJpRzYiSSJqR0Yl.
<Text-field style="Heading 3" layout="Heading 3" bookmark="RegularDifferentialChains">Regular Differential Chains</Text-field>
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 NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU= with rational numbers for coefficients. Each differential polynomial NiMmSSJBRzYiNiNJImlHRiU= admits a leading derivative, NiMmSSJ2RzYiNiNJImlHRiU= which is a derivative. The list NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU= is sorted by increasing leading rank.
The set NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU= is differentially triangular and partially autoreduced:
Given any two different indices NiRJImlHNiJJImpHRiQ=, the leading derivative NiMmSSJ2RzYiNiNJImlHRiU= is not a derivative of NiMmSSJ2RzYiNiNJImpHRiU=.
Given any two indices NiRJImlHNiJJImpHRiQ=, the differential polynomial NiMmSSJBRzYiNiNJImlHRiU= does not depend on any proper derivative of NiMmSSJ2RzYiNiNJImlHRiU=.
At this stage, one needs to introduce the polynomial ring NiMmSSJSRzYiNiMiIiE= = NiNJImtHNiI= [NiMmSSJ2RzYiNiMiIiI=, ..., NiMmSSJ2RzYiNiNJInJHRiU=], which is obtained by moving into the base field, all the independent variables and derivatives, the NiMmSSJBRzYiNiNJImlHRiU= depend on, but which are not the leading derivative of any NiMmSSJBRzYiNiNJImlHRiU=.
The set NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU= is a squarefree regular chain:
Given any index NiNJImlHNiI=, the initial of NiMmSSJBRzYiNiNJImlHRiU= is an invertible element of NiMmSSJSRzYiNiMiIiE=/NiNeIyIiIg== where NiNeIyIiIg== denotes the ideal generated by NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiMsJkkiaUdGJSIiIiEiIkYp in the ring NiMmSSJSRzYiNiMiIiE=.
Given any index NiNJImlHNiI=, the separant of NiMmSSJBRzYiNiNJImlHRiU= is an invertible element of NiMmSSJSRzYiNiMiIiE=/NiNeIyIiIg== where NiNeIyIiIg== denotes the ideal generated by NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJImlHRiU= in the ring NiMmSSJSRzYiNiMiIiE=.
At this stage, the polynomial system NiMmSSJBRzYiNiNJInZHRiU= must be introduced, where NiNJInZHNiI= is a derivative. It is the set of all the derivatives of the differential polynomials NiMmSSJBRzYiNiNJImlHRiU= whose leading derivatives are strictly less than NiNJInZHNiI=. The ring NiMmSSJSRzYiNiNJInZHRiU= 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 NiMmSSJBRzYiNiNJInZHRiU= depend on, but which are not the leading derivative of any element of NiMmSSJBRzYiNiNJInZHRiU=.
In the case of partial differential systems, the set NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU= is coherent: given any two different indices NiRJImlHNiJJImpHRiQ=, such that NiMmSSJ2RzYiNiNJImlHRiU= and NiMmSSJ2RzYiNiNJImpHRiU= are derivatives of some same dependent variable NiNJInVHNiI=, the NiMtSSNtaUc2JEkqcHJvdGVjdGVkR0YmL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJkkoX3N5c2xpYkdGKTYjUSgmRGVsdGE7Rik=-polynomial defined by NiMmSSJBRzYiNiNJImlHRiU= and NiMmSSJBRzYiNiNJImpHRiU= is zero in the ring NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EiUkYpLUklbXJvd0c2JEYmL0YoRio2JS1GJDYkLUYvNiNRKCZ0aGV0YTtGKS1GNTYlLUYvNiNRImlGKS1JI21vRzYkRiYvRihGKjYjUSgmY29tbWE7RiktRi82I1EiakYpLUZENiNRMSZJbnZpc2libGVUaW1lcztGKS1GLzYjUSJ1Rik=/NiNeIyIiIg== where NiMtSSVtcm93RzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiUtSSVtc3ViRzYkRiYvRihGKjYkLUkjbWlHNiRGJi9GKEYqNiNRKCZ0aGV0YTtGKS1GJDYlLUY0NiNRImlGKS1JI21vRzYkRiYvRihGKjYjUSgmY29tbWE7RiktRjQ2I1EiakYpLUY/NiNRMSZJbnZpc2libGVUaW1lcztGKS1GNDYjUSJ1Rik= is least common derivative of NiMmSSJ2RzYiNiNJImlHRiU= and NiMmSSJ2RzYiNiNJImpHRiU= and NiNeIyIiIg== denotes the ideal generated by NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EiQUYpLUklbXJvd0c2JEYmL0YoRio2JS1GJDYkLUYvNiNRKCZ0aGV0YTtGKS1GNTYlLUYvNiNRImlGKS1JI21vRzYkRiYvRihGKjYjUSgmY29tbWE7RiktRi82I1EiakYpLUZENiNRMSZJbnZpc2libGVUaW1lcztGKS1GLzYjUSJ1Rik= in NiMtSSVtc3ViRzYkSSpwcm90ZWN0ZWRHRiYvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSNtaUc2JEYmL0YoRio2I1EiUkYpLUklbXJvd0c2JEYmL0YoRio2JS1GJDYkLUYvNiNRKCZ0aGV0YTtGKS1GNTYlLUYvNiNRImlGKS1JI21vRzYkRiYvRihGKjYjUSgmY29tbWE7RiktRi82I1EiakYpLUZENiNRMSZJbnZpc2libGVUaW1lcztGKS1GLzYjUSJ1Rik=.
<Text-field style="Heading 3" layout="Heading 3" bookmark="DifferentialIdeals">Differential Ideals</Text-field>
Every regular differential chain NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU=, of a polynomial differential ring NiNJIlJHNiI=, defines a differential ideal NiNeIyIiIg==, which is the set of all the differential polynomials NiNJIkZHNiI= such that, for some power product NiNJIkhHNiI= of the initials and the separants of the NiMmSSJBRzYiNiNJImlHRiU=, the differential polynomial NiMqJkkiSEc2IiIiIkkiRkdGJUYm is a finite linear combination of the derivatives of the NiMmSSJBRzYiNiNJImlHRiU=, with differential polynomials of NiNJIlJHNiI= for coefficients.
One stresses the fact the chain NiMmSSJBRzYiNiMiIiI=, ..., NiMmSSJBRzYiNiNJInJHRiU= is not a generating family of NiNeIyIiIg==. However, it completely defines NiNeIyIiIg== and permits to compute a normal form of the residue class of any differential polynomial in NiNJIlJHNiI=/NiNeIyIiIg==. See NormalForm. In particular:
It permits to decide membership in NiNeIyIiIg==, that is, zero in NiNJIlJHNiI=/NiNeIyIiIg==: a differential polynomial NiNJIkZHNiI= belongs to NiNeIyIiIg== if and only if its normal form is NiMiIiE=.
It permits to decide if a differential polynomial NiNJIkZHNiI= is regular modulo NiNeIyIiIg==, that is, a non-zero divisor in NiNJIlJHNiI=/NiNeIyIiIg==.
Given any system of differential polynomials NiMmSSJGRzYiNiMiIiI=, ..., NiMmSSJGRzYiNiNJInNHRiU= and any ranking, it is possible to compute a representation of the radical NiNJIkpHNiI= of the differential ideal generated by this system, as a finite intersection of radical differential ideals NiMmXiMiIiI2I0Yl, ..., NiMmXiMiIiI2I0kibkc2Ig==, presented by regular differential chains NiMmSSJDRzYiNiMiIiI=, ..., NiMmSSJDRzYiNiNJIm5HRiU=. See RosenfeldGroebner [BLOP95,BLOP09]. See also [W98,LW99,H00,BKM01]. This representation permits to decide membership in NiNJIkpHNiI=: a differential polynomial NiNJIkZHNiI= belongs to NiNJIkpHNiI= if and only if it belongs to each differential ideal NiMmXiMiIiI2I0kia0c2Ig==, i.e., if and only if its normal forms, with respect to all the regular differential chains NiMmSSJDRzYiNiNJImtHRiU=, are all NiMiIiE=. See BelongsTo.
One stresses the fact that, in general, the computed representation NiMmSSJDRzYiNiMiIiI=, ..., NiMmSSJDRzYiNiNJIm5HRiU= 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 NiNJIlFHNiI= (NiMmSSJ4RzYiNiMiIiI=, ..., NiMmSSJ4RzYiNiNJInBHRiU=) 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.
<Text-field style="Heading 3" layout="Heading 3" bookmark="bkmrk2">Normal Forms</Text-field>
Let NiNJIkNHNiI= be a regular differential chain of a differential polynomial ring NiNJIlJHNiI= and NiNJIkZHNiI= be a differential polynomial of NiNJIlJHNiI=. Let NiNeIyIiIg== be the differential ideal defined by NiNJIkNHNiI=. The normal form of NiNJIkZHNiI= with respect to NiNJIkNHNiI= is a rational differential fraction NiNJIlBHNiI=/NiNJIlFHNiI= such that
NiNJIlFHNiI= is regular (that is, a non-zero divisor) in NiNJIlJHNiI=/NiNeIyIiIg==.
NiNJIlBHNiI=/NiNJIlFHNiI= is equivalent to NiNJIkZHNiI= modulo NiNeIyIiIg==, in the sense that NiMsJiomSSJGRzYiIiIiSSJRR0YmRidGJ0kiUEdGJiEiIg== belongs to NiNeIyIiIg==.
NiNJIlBHNiI= is reduced with respect to NiNJIkNHNiI=, in the sense that, given any leading rank NiMpSSJ2RzYiSSJkR0Yl of any element of NiNJIkNHNiI=, it does not depend on any proper derivative of NiNJInZHNiI=, and, has degree less than NiNJImRHNiI=, in NiNJInZHNiI=.
NiNJIlFHNiI= does not depend on any derivative of NiNJInZHNiI=, where NiNJInZHNiI= denotes any leading derivative of any element of NiNJIkNHNiI=.
The normal form NiNJIlBHNiI=/NiNJIlFHNiI= of NiNJIkZHNiI= with respect to NiNJIkNHNiI= is a canonical representative of the residue class of NiNJIkZHNiI= in NiNJIlJHNiI=/NiNeIyIiIg==, in the sense that
NiNJIlBHNiI=/NiNJIlFHNiI= is NiMiIiE= if and only if NiNJIkZHNiI= belongs to NiNeIyIiIg==.
If NiNJIkZHNiI= is equivalent to NiNJIkhHNiI= modulo NiNeIyIiIg==, then the normal forms of NiNJIkZHNiI= and NiNJIkhHNiI=, with respect to NiNJIkNHNiI=, are equal.
The normal form of NiNJIkZHNiI= can be computed by means of the NormalForm function [BL00].
Let us extend the above definition and consider a rational differential fraction NiNJIkZHNiI=/NiNJIkdHNiI=, where NiNJIkZHNiI= and NiNJIkdHNiI= are differential polynomials of NiNJIlJHNiI=. The NormalForm function can be used to compute a normal form of NiNJIkZHNiI=/NiNJIkdHNiI=, with respect to NiNJIkNHNiI=. If it succeeds, then it returns a rational differential fraction NiNJIlBHNiI=/NiNJIlFHNiI= such that
NiNJIlFHNiI= is regular (that is, a non-zero divisor) in NiNJIlJHNiI=/NiNeIyIiIg==.
NiNJIlBHNiI=/NiNJIlFHNiI= is equivalent to NiNJIkZHNiI=/NiNJIkdHNiI= modulo NiNeIyIiIg==, in the sense that NiMsJiomSSJGRzYiIiIiSSJRR0YmRidGJyomSSJHR0YmRidJIlBHRiZGJyEiIg== belongs to NiNeIyIiIg==.
NiNJIlBHNiI= is reduced with respect to NiNJIkNHNiI=.
NiNJIlFHNiI= does not depend on any derivative of any leading derivative of NiNJIkNHNiI=.
If it succeeds, then NiNJIkdHNiI= is regular in NiNJIlJHNiI=/NiNeIyIiIg==, and the normal form NiNJIlBHNiI=/NiNJIlFHNiI= is a canonical representative of the residue class of NiNJIkZHNiI=/NiNJIkdHNiI= in the total fraction ring of NiNJIlJHNiI=/NiNeIyIiIg==, in the sense that
NiNJIlBHNiI=/NiNJIlFHNiI= is NiMiIiE= if and only if NiNJIkZHNiI= belongs to NiNeIyIiIg==.
If NiNJIkZHNiI=/NiNJIkdHNiI= is equivalent to NiNJIkhHNiI=/NiNJIktHNiI= modulo NiNeIyIiIg== (with NiNJIktHNiI= regular modulo NiNeIyIiIg==), then the normal forms of NiNJIkZHNiI=/NiNJIkdHNiI= and NiNJIkhHNiI=/NiNJIktHNiI= with respect to NiNJIkNHNiI= are equal.
The NormalForm function may fail to compute a normal form of NiNJIkZHNiI=/NiNJIkdHNiI= in the following cases:
If NiNJIkdHNiI= is zero in NiNJIlJHNiI=/NiNeIyIiIg==.
If NiNJIkdHNiI= is a zero-divisor in NiNJIlJHNiI=/NiNeIyIiIg==.
If the function is led to invert another zero-divisor in NiNJIlJHNiI=/NiNeIyIiIg==, during the normal form computation.
However, for each rational differential fraction NiNJIkZHNiI=/NiNJIkdHNiI=, it is possible to split NiNJIkNHNiI= into finitely many regular differential chains NiMmSSJDRzYiNiMiIiI=, ..., NiMmSSJDRzYiNiNJIm5HRiU=, NiMmSSJDRzYiNiMsJkkibkdGJSIiIkYpRik=, ..., NiMmSSJDRzYiNiMsJkkibkdGJSIiIkkicEdGJUYp such that, denoting NiMmXiMiIiI2I0kia0c2Ig== the radical differential ideal defined by NiMmSSJDRzYiNiNJImtHRiU=
The normal form of NiNJIkZHNiI=/NiNJIkdHNiI= can be computed, with respect to NiMmSSJDRzYiNiNJImtHRiU=, for NiMiIiI= <= k <= NiNJIm5HNiI=.
NiNJIkdHNiI= is zero in NiNJIlJHNiI=/NiMmXiMiIiI2I0kia0c2Ig==, for NiMsJkkibkc2IiIiIkYmRiY= <= NiNJImtHNiI= <= NiMsJkkibkc2IiIiIkkicEdGJUYm.
The differential ideal NiNeIyIiIg== is equal to the intersection of the differential ideals NiMmXiMiIiI2I0kia0c2Ig==, for NiMiIiI= <= k <= NiMsJkkibkc2IiIiIkkicEdGJUYm,
This decomposition can be achieved by using the NormalForm function.
See AlsoDifferentialAlgebra