Basic Operations
(1) -> )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 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, %) -> % ?*? : (%, Integer) -> % if and(has(R, LinearlyExplicitOver(Integer)), has(R, Ring)) ?*? : (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 hash : % -> SingleInteger if V has HASHABL and R has HASHABL hashUpdate! : (HashState, %) -> HashState if V has HASHABL and R has HASHABL 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 plenaryPower : (%, PositiveInteger) -> % if R has GCDDOM or R has COMRING and % has ATVCWC or R has INTDOM and % has ATVCWC or R has ALGEBRA(FRAC(INT)) 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)
Kolchin, E.R. "Differential Algebra and Algebraic Groups" (Academic Press, 1973)
Ritt, J.F. "Differential Algebra" (Dover, 1950).
Example
odvar:=ODVAR Symbol
-- here are the first 5 derivatives of w
-- the i-th derivative of w is printed as w subscript 5
[makeVariable('w,
-- these are now algebraic indeterminates,ranked in an orderly way -- in increasing order: sort %
-- 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);
-- 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)$dpol
-- create another one called z,which is higher in rank than w -- since we are ordering by Symbol z := makeVariable('z)$dpol
-- now define some differential polynomial (f,b):dpol
f:=w.4::dpol - w.1 * w.1 * z.3
b:=(z.1::dpol)^3 * (z.2)^2 - w.2
-- compute the leading derivative appearing in b lb:=leader b
-- the separant is the partial derivative of b with respect to its leader sb:=separant b
-- 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
-- find its leader lbprime:= leader bprime
-- differentiate f partially with respect to lbprime pbf:=differentiate (f,lbprime)
-- to obtain the partial remainder of f with respect to b ftilde:=sb * f- pbf * bprime
-- 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
-- compute the leading coefficient of ftilde -- as a polynomial in its leader lcef:=leadingCoefficient univariate(ftilde,lb)
-- 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