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 $p$ and $q$ of the chain, the degree of $p$ in the leading derivative of $q$ is less than the degree of $q$ 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 $\mathrm{\Delta }$-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 $S$-polynomials of the Groebner bases theory. Let $A_i$ and $A_j$ be two differential polynomials, whose leading derivatives $v_i$ and $v_j$ are derivatives of some same dependent variable $u$. Denote ${\mathrm{\theta }}_i$ and ${\mathrm{\theta }}_j$ the derivation operators associated to $v_i$ and $v_j$ and ${\mathrm{\theta }}_{i,j}$ their least common multiple.

If ${\mathrm{\theta }}_{i,j}$ is different from both ${\mathrm{\theta }}_i$ and ${\mathrm{\theta }}_j$, then the $\mathrm{\Delta }$-polynomial defined by $A_i$ and $A_j$ is the differential polynomial $S_i$ $\frac{{\mathrm{\theta }}_{i,j}}{{\mathrm{\theta }}_j}$ $A_j$ - $S_j$ $\frac{{\mathrm{\theta }}_{i,j}}{{\mathrm{\theta }}_i}$ $A_i$, where $S_i$ and $S_j$ denote the separants of $A_i$ and $A_j$ and the left multiplication by a derivation operator stands for a differentiation.

If ${\mathrm{\theta }}_{i,j}$ is equal to ${\mathrm{\theta }}_j$ (as an example), then the $\mathrm{\Delta }$-polynomial defined by $A_i$ and $A_j$ is the pseudo-remainder of $A_j$ by $\frac{{\mathrm{\theta }}_j}{{\mathrm{\theta }}_i}$ $A_i$ with respect to $v_j$.

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 $v$ is a derivative of a dependent variable $u$, then $u$ is said to be the dependent variable associated to $v$.

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 $p$, with respect to the derivation operator $x^{2}y$, consists of differentiating $p$ twice with respect to $x$, then once with respect to $y$. The identity derivation operator is denoted as $1$.

derivation operator associated to a derivative: If $v$ is a derivative of a dependent variable $u$, then there exists a derivation operator, $\mathrm{\theta }$, such that differentiating $u$ with respect to $\mathrm{\theta }$ gives $v$. The derivation operator $\mathrm{\theta }$ is called the derivation operator associated to $v$.

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 $p$, whenever it contains $p$. 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 $t$, and the space variables $x$, $y$, $z$.

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

irredundant: A representation of an ideal $J$ as an intersection of ideals ${\mathrm{I}}_1$, ..., ${\mathrm{I}}_n$ is said to be irredundant if, given any two different indices $j,k$ the ideal ${\mathrm{I}}_j$ is not included in the ideal ${\mathrm{I}}_k$. In general, the representations computed by the RosenfeldGroebner algorithm are redundant. This issue is addressed in a following section.

leading derivative: The leading derivative $v$ of a non-numeric differential polynomial $p$, is the highest derivative, with respect to some given ranking, such that the degree of $p$ in $v$ 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 $p$ is the rank $v^d$ such that $v$ is the leading derivative of $p$, and, $d$ is the degree of $p$ in $v$. In this package, the leading rank of differential polynomials which do not depend on any derivative is defined. In particular, the leading rank of $0$ is $0$, the one of any other rational number is $1$.

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 $u$ and $v$, $u$ is higher than $v$ whenever the order  of $u$ is greater than the order of $v$.

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 $pq$, 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 $1$. 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 $v$ is said to be a proper derivative of a derivative $u$, if $v$ is a derivative of $u$ and is different from $u$.

radical: An ideal $\mathrm{I}$ of a ring $R$ is said to be radical if it involves some element $p$ whenever it involves any power $p^d$ of $p$, where $d$ is any non-negative integer. The radical of an ideal $\mathrm{I}$ is the set of all the elements $p$ of $R$, a power of which belongs to $\mathrm{I}$. 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 $0$ and $1$ are defined. Any ranking extends to a total ordering on ranks: $0$ is less than $1$, which is less than any other rank. Two ranks $u^n$ and $v^m$ are compared by comparing, first, $u$ and $v$, then, the exponents $n$ and $m$.

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 $\mathrm{I}$ is an ideal of a ring $R$ and $M$ is a multiplicative family of $R$, then the saturation of $\mathrm{I}$ by $M$ is the set $J$ of all the elements $p$ of $R$ such that, $mp$ belongs to $\mathrm{I}$, for some $m$ of $M$. 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 $p$, is the partial derivative of $p$ 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 $p$, is equal to $p-c{v}^d$ where $c$ is the initial of $p$ and $v^d$ is its leading rank.

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