MathAction SandBoxDifferentialPolynomial

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Description: DifferentialSparseMultivariatePolynomial? implements an ordinary differential polynomial ring by combining a domain belonging to the category DifferentialVariableCategory? with the domain SparseMultivariatePolynomial?.
Author: William Sit
Date Created: 19 July 1990
Date Last Updated: 13 September 1991

Basic Operations

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)show DifferentialPolynomialCategory

 DifferentialPolynomialCategory(R: Ring,S: OrderedSet,V: DifferentialVariableCategory(t#2),E: OrderedAbelianMonoidSup) is a category constructor
 Abbreviation for DifferentialPolynomialCategory is DPOLCAT 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------

 ?*? : (Integer, %) -> %               ?*? : (R, %) -> %
 ?*? : (%, R) -> %                     ?*? : (%, %) -> %
 ?*? : (PositiveInteger, %) -> %       ?+? : (%, %) -> %
 ?-? : (%, %) -> %                     -? : % -> %
 ?=? : (%, %) -> Boolean               D : (%, (R -> R)) -> %
 D : % -> % if R has DIFRING           1 : () -> %
 0 : () -> %                           ?^? : (%, PositiveInteger) -> %
 annihilate? : (%, %) -> Boolean       antiCommutator : (%, %) -> %
 associator : (%, %, %) -> %           coefficient : (%, E) -> R
 coefficients : % -> List(R)           coerce : S -> %
 coerce : Integer -> %                 coerce : R -> %
 coerce : % -> OutputForm              commutator : (%, %) -> %
 degree : % -> E                       eval : (%, V, %) -> %
 eval : (%, V, R) -> %                 ground : % -> R
 ground? : % -> Boolean                hash : % -> SingleInteger
 initial : % -> %                      isobaric? : % -> Boolean
 latex : % -> String                   leader : % -> V
 map : ((R -> R), %) -> %              minimumDegree : % -> E
 monomial : (R, E) -> %                monomial? : % -> Boolean
 monomials : % -> List(%)              one? : % -> Boolean
 opposite? : (%, %) -> Boolean         order : % -> NonNegativeInteger
 pomopo! : (%, R, E, %) -> %           recip : % -> Union(%,"failed")
 retract : % -> S                      retract : % -> R
 sample : () -> %                      separant : % -> %
 support : % -> List(E)                variables : % -> List(V)
 weight : % -> NonNegativeInteger      zero? : % -> Boolean
 ?~=? : (%, %) -> Boolean             
 ?*? : (NonNegativeInteger, %) -> %
 ?*? : (%, Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT))
 ?*? : (Fraction(Integer), %) -> % if R has ALGEBRA(FRAC(INT))
 ?/? : (%, R) -> % if R has FIELD
 D : (%, (R -> R), NonNegativeInteger) -> %
 D : (%, List(Symbol), List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL)
 D : (%, Symbol, NonNegativeInteger) -> % if R has PDRING(SYMBOL)
 D : (%, List(Symbol)) -> % if R has PDRING(SYMBOL)
 D : (%, Symbol) -> % if R has PDRING(SYMBOL)
 D : (%, NonNegativeInteger) -> % if R has DIFRING
 D : (%, V) -> % if R has RING
 D : (%, List(V)) -> % if R has RING
 D : (%, V, NonNegativeInteger) -> % if R has RING
 D : (%, List(V), List(NonNegativeInteger)) -> % if R has RING
 ?^? : (%, NonNegativeInteger) -> %
 associates? : (%, %) -> Boolean if R has ENTIRER or R has INTDOM and % has ATVCWC
 binomThmExpt : (%, %, NonNegativeInteger) -> % if % has COMRING
 characteristic : () -> NonNegativeInteger
 charthRoot : % -> Union(%,"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) or R has CHARNZ
 coefficient : (%, List(V), List(NonNegativeInteger)) -> %
 coefficient : (%, V, NonNegativeInteger) -> %
 coerce : V -> % if R has SRING
 coerce : % -> % if R has GCDDOM or R has COMRING and % has ATVCWC or R has INTDOM and % has ATVCWC
 coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT))
 conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit))
 construct : List(Record(k: E,c: R)) -> %
 constructOrdered : List(Record(k: E,c: R)) -> % if E has COMPAR
 content : (%, V) -> % if R has GCDDOM
 content : % -> R if R has GCDDOM
 convert : % -> InputForm if V has KONVERT(INFORM) and R has KONVERT(INFORM)
 convert : % -> Pattern(Integer) if V has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) and R has RING
 convert : % -> Pattern(Float) if V has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) and R has RING
 degree : (%, S) -> NonNegativeInteger
 degree : (%, List(V)) -> List(NonNegativeInteger)
 degree : (%, V) -> NonNegativeInteger
 differentialVariables : % -> List(S)
 differentiate : (%, (R -> R)) -> %
 differentiate : (%, (R -> R), NonNegativeInteger) -> %
 differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL)
 differentiate : (%, Symbol, NonNegativeInteger) -> % if R has PDRING(SYMBOL)
 differentiate : (%, List(Symbol)) -> % if R has PDRING(SYMBOL)
 differentiate : (%, Symbol) -> % if R has PDRING(SYMBOL)
 differentiate : (%, NonNegativeInteger) -> % if R has DIFRING
 differentiate : % -> % if R has DIFRING
 differentiate : (%, V) -> % if R has RING
 differentiate : (%, List(V)) -> % if R has RING
 differentiate : (%, V, NonNegativeInteger) -> % if R has RING
 differentiate : (%, List(V), List(NonNegativeInteger)) -> % if R has RING
 discriminant : (%, V) -> % if R has COMRING
 eval : (%, List(S), List(R)) -> % if R has DIFRING
 eval : (%, S, R) -> % if R has DIFRING
 eval : (%, List(S), List(%)) -> % if R has DIFRING
 eval : (%, S, %) -> % if R has DIFRING
 eval : (%, List(%), List(%)) -> % if R has SRING
 eval : (%, %, %) -> % if R has SRING
 eval : (%, Equation(%)) -> % if R has SRING
 eval : (%, List(Equation(%))) -> % if R has SRING
 eval : (%, List(V), List(%)) -> %
 eval : (%, List(V), List(R)) -> %
 exquo : (%, %) -> Union(%,"failed") if R has ENTIRER or R has INTDOM and % has ATVCWC
 exquo : (%, R) -> Union(%,"failed") if R has ENTIRER
 factor : % -> Factored(%) if R has PFECAT
 factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT
 factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT
 fmecg : (%, E, R, %) -> % if R has RING
 gcd : (%, %) -> % if R has GCDDOM
 gcd : List(%) -> % if R has GCDDOM
 gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM
 hashUpdate! : (HashState, %) -> HashState
 isExpt : % -> Union(Record(var: V,exponent: NonNegativeInteger),"failed") if R has SRING
 isPlus : % -> Union(List(%),"failed")
 isTimes : % -> Union(List(%),"failed") if R has SRING
 lcm : (%, %) -> % if R has GCDDOM
 lcm : List(%) -> % if R has GCDDOM
 lcmCoef : (%, %) -> Record(llcm_res: %,coeff1: %,coeff2: %) if R has GCDDOM
 leadingCoefficient : % -> R if E has COMPAR
 leadingMonomial : % -> % if E has COMPAR
 leadingSupport : % -> E if E has COMPAR
 leadingTerm : % -> Record(k: E,c: R) if E has COMPAR
 leftPower : (%, NonNegativeInteger) -> %
 leftPower : (%, PositiveInteger) -> %
 leftRecip : % -> Union(%,"failed")
 linearExtend : ((E -> R), %) -> R if R has COMRING
 listOfTerms : % -> List(Record(k: E,c: R))
 mainVariable : % -> Union(V,"failed")
 makeVariable : % -> (NonNegativeInteger -> %) if R has DIFRING
 makeVariable : S -> (NonNegativeInteger -> %)
 mapExponents : ((E -> E), %) -> %
 minimumDegree : (%, List(V)) -> List(NonNegativeInteger)
 minimumDegree : (%, V) -> NonNegativeInteger
 monicDivide : (%, %, V) -> Record(quotient: %,remainder: %) if R has RING
 monomial : (%, List(V), List(NonNegativeInteger)) -> %
 monomial : (%, V, NonNegativeInteger) -> %
 multivariate : (SparseUnivariatePolynomial(%), V) -> %
 multivariate : (SparseUnivariatePolynomial(R), V) -> %
 numberOfMonomials : % -> NonNegativeInteger
 order : (%, S) -> NonNegativeInteger
 patternMatch : (%, Pattern(Integer), PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if V has PATMAB(INT) and R has PATMAB(INT) and R has RING
 patternMatch : (%, Pattern(Float), PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if V has PATMAB(FLOAT) and R has PATMAB(FLOAT) and R has RING
 prime? : % -> Boolean if R has PFECAT
 primitiveMonomials : % -> List(%) if R has SRING
 primitivePart : (%, V) -> % if R has GCDDOM
 primitivePart : % -> % if R has GCDDOM
 reducedSystem : Matrix(%) -> Matrix(R) if R has RING
 reducedSystem : (Matrix(%), Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) if R has RING
 reducedSystem : (Matrix(%), Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if and(has(R,LinearlyExplicitOver(Integer)),has(R,Ring))
 reducedSystem : Matrix(%) -> Matrix(Integer) if and(has(R,LinearlyExplicitOver(Integer)),has(R,Ring))
 reductum : % -> % if E has COMPAR
 resultant : (%, %, V) -> % if R has COMRING
 retract : % -> V if R has SRING
 retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT))
 retract : % -> Integer if R has RETRACT(INT)
 retractIfCan : % -> Union(S,"failed")
 retractIfCan : % -> Union(V,"failed") if R has SRING
 retractIfCan : % -> Union(R,"failed")
 retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT))
 retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT)
 rightPower : (%, NonNegativeInteger) -> %
 rightPower : (%, PositiveInteger) -> %
 rightRecip : % -> Union(%,"failed")
 smaller? : (%, %) -> Boolean if R has COMPAR
 solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)), SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has PFECAT
 squareFree : % -> Factored(%) if R has GCDDOM
 squareFreePart : % -> % if R has GCDDOM
 squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT
 subtractIfCan : (%, %) -> Union(%,"failed")
 totalDegree : (%, List(V)) -> NonNegativeInteger
 totalDegree : % -> NonNegativeInteger
 totalDegreeSorted : (%, List(V)) -> NonNegativeInteger
 unit? : % -> Boolean if R has ENTIRER or R has INTDOM and % has ATVCWC
 unitCanonical : % -> % if R has ENTIRER or R has INTDOM and % has ATVCWC
 unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has ENTIRER or R has INTDOM and % has ATVCWC
 univariate : % -> SparseUnivariatePolynomial(R)
 univariate : (%, V) -> SparseUnivariatePolynomial(%)
 weight : (%, S) -> NonNegativeInteger
 weights : (%, S) -> List(NonNegativeInteger)
 weights : % -> List(NonNegativeInteger)

References

http://en.wikipedia.org/wiki/Differential_algebra

Kolchin, E.R. "Differential Algebra and Algebraic Groups" (Academic Press, 1973)

Ritt, J.F. "Differential Algebra" (Dover, 1950).

Example

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odvar:=ODVAR Symbol

$\label{eq1}\hbox{\axiomType{OrderlyDifferentialVariable}\ } (\hbox{\axiomType{Symbol}\ })$

(1)

Type: Type

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-- here are the first 5 derivatives of w
-- the i-th derivative of w is printed as w subscript 5
[makeVariable('w,i)$odvar for i in 5..0 by -1]

$\label{eq2}\left[{w_{5}}, \:{w_{4}}, \:{w_{3}}, \:{w_{2}}, \:{w_{1}}, \: w \right]$

(2)

Type: List(OrderlyDifferentialVariable?(Symbol))

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-- these are now algebraic indeterminates, ranked in an orderly way
-- in increasing order:
sort %

$\label{eq3}\left[ w , \:{w_{1}}, \:{w_{2}}, \:{w_{3}}, \:{w_{4}}, \:{w_{5}}\right]$

(3)

Type: List(OrderlyDifferentialVariable?(Symbol))

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-- we now make a general differential polynomial ring
-- instead of ODVAR, one can also use SDVAR for sequential ordering
dpol:=DSMP (FRAC INT, Symbol, odvar);

Type: Type

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-- instead of using makeVariable, it is easier to
-- think of a differential variable w as a map, where
-- w.n is n-th derivative of w as an algebraic indeterminate
w := makeVariable('w)$dpol

$\label{eq4}\mbox{theMap (...)}$

(4)

Type: (NonNegativeInteger? -> DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol)))

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-- create another one called z, which is higher in rank than w
-- since we are ordering by Symbol
z := makeVariable('z)$dpol

$\label{eq5}\mbox{theMap (...)}$

(5)

Type: (NonNegativeInteger? -> DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol)))

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-- now define some differential polynomial
(f,b):dpol

Type: Void

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f:=w.4::dpol - w.1 * w.1 * z.3

$\label{eq6}{w_{4}}-{{{w_{1}}^{2}}\ {z_{3}}}$

(6)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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b:=(z.1::dpol)^3 * (z.2)^2 - w.2

$\label{eq7}{{{z_{1}}^{3}}\ {{z_{2}}^{2}}}-{w_{2}}$

(7)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- compute the leading derivative appearing in b
lb:=leader b

$\label{eq8}z_{2}$

(8)

Type: OrderlyDifferentialVariable?(Symbol)

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-- the separant is the partial derivative of b with respect to its leader
sb:=separant b

$\label{eq9}2 \ {{z_{1}}^{3}}\ {z_{2}}$

(9)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- of course you can differentiate these differential polynomials
-- and try to reduce f modulo the differential ideal generated by b
-- first eliminate z.3 using the derivative of b
bprime:= differentiate b

$\label{eq10}{2 \ {{z_{1}}^{3}}\ {z_{2}}\ {z_{3}}}-{w_{3}}+{3 \ {{z_{1}}^{2}}\ {{z_{2}}^{3}}}$

(10)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- find its leader
lbprime:= leader bprime

$\label{eq11}z_{3}$

(11)

Type: OrderlyDifferentialVariable?(Symbol)

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-- differentiate f partially with respect to lbprime
pbf:=differentiate (f, lbprime)

$\label{eq12}-{{w_{1}}^{2}}$

(12)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- to obtain the partial remainder of f with respect to b
ftilde:=sb * f- pbf * bprime

$\label{eq13}{2 \ {{z_{1}}^{3}}\ {z_{2}}\ {w_{4}}}-{{{w_{1}}^{2}}\ {w_{3}}}+{3 \ {{w_{1}}^{2}}\ {{z_{1}}^{2}}\ {{z_{2}}^{3}}}$

(13)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- note high powers of lb still appears in ftilde
-- the initial is the leading coefficient when b is written
-- as a univariate polynomial in its leader
ib:=initial b

$\label{eq14}{z_{1}}^{3}$

(14)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- compute the leading coefficient of ftilde
-- as a polynomial in its leader
lcef:=leadingCoefficient univariate(ftilde, lb)

$\label{eq15}3 \ {{w_{1}}^{2}}\ {{z_{1}}^{2}}$

(15)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

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-- now to continue eliminating the high powers of lb appearing in ftilde:
-- to obtain the remainder of f modulo b and its derivatives

f0:=ib * ftilde - lcef * b * lb

$\label{eq16}{2 \ {{z_{1}}^{6}}\ {z_{2}}\ {w_{4}}}-{{{w_{1}}^{2}}\ {{z_{1}}^{3}}\ {w_{3}}}+{3 \ {{w_{1}}^{2}}\ {{z_{1}}^{2}}\ {w_{2}}\ {z_{2}}}$

(16)

Type: DifferentialSparseMultivariatePolynomial?(Fraction(Integer),Symbol,OrderlyDifferentialVariable?(Symbol))

Subject: Be Bold !!

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