\documentclass{article} \usepackage{axiom} \begin{document} \title{\$SPAD/src/algebra polycat.spad} \author{Manuel Bronstein} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{category AMR AbelianMonoidRing} <>= )abbrev category AMR AbelianMonoidRing ++ Author: ++ Date Created: ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ Abelian monoid ring elements (not necessarily of finite support) ++ of this ring are of the form formal SUM (r_i * e_i) ++ where the r_i are coefficents and the e_i, elements of the ++ ordered abelian monoid, are thought of as exponents or monomials. ++ The monomials commute with each other, and with ++ the coefficients (which themselves may or may not be commutative). ++ See \spadtype{FiniteAbelianMonoidRing} for the case of finite support ++ a useful common model for polynomials and power series. ++ Conceptually at least, only the non-zero terms are ever operated on. AbelianMonoidRing(R:Ring, E:OrderedAbelianMonoid): Category == Join(Ring,BiModule(R,R)) with leadingCoefficient: % -> R ++ leadingCoefficient(p) returns the coefficient highest degree term of p. leadingMonomial: % -> % ++ leadingMonomial(p) returns the monomial of p with the highest degree. degree: % -> E ++ degree(p) returns the maximum of the exponents of the terms of p. map: (R -> R, %) -> % ++ map(fn,u) maps function fn onto the coefficients ++ of the non-zero monomials of u. monomial?: % -> Boolean ++ monomial?(p) tests if p is a single monomial. monomial: (R,E) -> % ++ monomial(r,e) makes a term from a coefficient r and an exponent e. reductum: % -> % ++ reductum(u) returns u minus its leading monomial ++ returns zero if handed the zero element. coefficient: (%,E) -> R ++ coefficient(p,e) extracts the coefficient of the monomial with ++ exponent e from polynomial p, or returns zero if exponent is not present. if R has Field then "/": (%,R) -> % ++ p/c divides p by the coefficient c. if R has CommutativeRing then CommutativeRing Algebra R if R has CharacteristicZero then CharacteristicZero if R has CharacteristicNonZero then CharacteristicNonZero if R has IntegralDomain then IntegralDomain if R has Algebra Fraction Integer then Algebra Fraction Integer add monomial? x == zero? reductum x map(fn:R -> R, x: %) == -- this default definition assumes that reductum is cheap zero? x => 0 r:=fn leadingCoefficient x zero? r => map(fn,reductum x) monomial(r, degree x) + map(fn,reductum x) if R has Algebra Fraction Integer then q:Fraction(Integer) * p:% == map(q * #1, p) @ \section{category FAMR FiniteAbelianMonoidRing} <>= import Boolean import NonNegativeInteger import List )abbrev category FAMR FiniteAbelianMonoidRing ++ Author: ++ Date Created: ++ Date Last Updated: 14.08.2000 Exported pomopo! and binomThmExpt [MMM] ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: This category is ++ similar to AbelianMonoidRing, except that the sum is assumed to be finite. ++ It is a useful model for polynomials, ++ but is somewhat more general. FiniteAbelianMonoidRing(R:Ring, E:OrderedAbelianMonoid): Category == Join(AbelianMonoidRing(R,E),FullyRetractableTo R) with ground?: % -> Boolean ++ ground?(p) tests if polynomial p is a member of the coefficient ring. -- can't be defined earlier, since a power series -- might not know if there were other terms or not ground: % -> R ++ ground(p) retracts polynomial p to the coefficient ring. coefficients: % -> List R ++ coefficients(p) gives the list of non-zero coefficients of polynomial p. numberOfMonomials: % -> NonNegativeInteger ++ numberOfMonomials(p) gives the number of non-zero monomials in polynomial p. minimumDegree: % -> E ++ minimumDegree(p) gives the least exponent of a non-zero term of polynomial p. ++ Error: if applied to 0. mapExponents: (E -> E, %) -> % ++ mapExponents(fn,u) maps function fn onto the exponents ++ of the non-zero monomials of polynomial u. pomopo!: (%,R,E,%) -> % ++ \spad{pomopo!(p1,r,e,p2)} returns \spad{p1 + monomial(e,r) * p2} ++ and may use \spad{p1} as workspace. The constaant \spad{r} is ++ assumed to be nonzero. if R has CommutativeRing then binomThmExpt: (%,%,NonNegativeInteger) -> % ++ \spad{binomThmExpt(p,q,n)} returns \spad{(x+y)^n} ++ by means of the binomial theorem trick. if R has IntegralDomain then "exquo": (%,R) -> Union(%,"failed") ++ exquo(p,r) returns the exact quotient of polynomial p by r, or "failed" ++ if none exists. if R has GcdDomain then content: % -> R ++ content(p) gives the gcd of the coefficients of polynomial p. primitivePart: % -> % ++ primitivePart(p) returns the unit normalized form of polynomial p ++ divided by the content of p. add pomopo!(p1,r,e,p2) == p1 + r * mapExponents(#1+e,p2) if R has CommutativeRing then binomThmExpt(x,y,nn) == nn = 0 => 1$% ans,xn,yn: % bincoef: Integer powl: List(%):= [x] i : Integer for i in 2..nn repeat powl:=[x * powl.first, :powl] yn:=y; ans:=powl.first; i:=1; bincoef:=nn for xn in powl.rest repeat ans:= bincoef * xn * yn + ans bincoef:= (nn-i) * bincoef quo (i+1); i:= i+1 -- last I and BINCOEF unused yn:= y * yn ans + yn ground? x == retractIfCan(x)@Union(R,"failed") case "failed" => false true ground x == retract(x)@R mapExponents (fn:E -> E, x: %) == -- this default definition assumes that reductum is cheap zero? x => 0 monomial(leadingCoefficient x,fn degree x)+mapExponents(fn,reductum x) coefficients x == zero? x => empty() concat(leadingCoefficient x, coefficients reductum x) if R has Field then x/r == map(#1/r,x) if R has IntegralDomain then (x: %) exquo (r: R) == -- probably not a very good definition in most special cases zero? x => 0 ans:% :=0 t:=leadingCoefficient x exquo r while not (t case "failed") and not zero? x repeat ans:=ans+monomial(t::R,degree x) x:=reductum x if not zero? x then t:=leadingCoefficient x exquo r t case "failed" => "failed" ans if R has GcdDomain then content x == -- this assumes reductum is cheap zero? x => 0 r:=leadingCoefficient x x:=reductum x while not zero? x and not one? r repeat r:=gcd(r,leadingCoefficient x) x:=reductum x r primitivePart x == zero? x => x c := content x unitCanonical((x exquo c)::%) @ \section{category POLYCAT PolynomialCategory} <>= )abbrev category POLYCAT PolynomialCategory ++ Author: ++ Date Created: ++ Date Last Updated: ++ Basic Functions: Ring, monomial, coefficient, differentiate, eval ++ Related Constructors: Polynomial, DistributedMultivariatePolynomial ++ Also See: UnivariatePolynomialCategory ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ The category for general multi-variate polynomials over a ring ++ R, in variables from VarSet, with exponents from the ++ \spadtype{OrderedAbelianMonoidSup}. PolynomialCategory(R:Ring, E:OrderedAbelianMonoidSup, VarSet:OrderedSet): Category == Join(PartialDifferentialRing VarSet, FiniteAbelianMonoidRing(R, E), Evalable %, InnerEvalable(VarSet, R), InnerEvalable(VarSet, %), RetractableTo VarSet, FullyLinearlyExplicitRingOver R) with -- operations degree : (%,VarSet) -> NonNegativeInteger ++ degree(p,v) gives the degree of polynomial p with respect to the variable v. degree : (%,List(VarSet)) -> List(NonNegativeInteger) ++ degree(p,lv) gives the list of degrees of polynomial p ++ with respect to each of the variables in the list lv. coefficient: (%,VarSet,NonNegativeInteger) -> % ++ coefficient(p,v,n) views the polynomial p as a univariate ++ polynomial in v and returns the coefficient of the \spad{v**n} term. coefficient: (%,List VarSet,List NonNegativeInteger) -> % ++ coefficient(p, lv, ln) views the polynomial p as a polynomial ++ in the variables of lv and returns the coefficient of the term ++ \spad{lv**ln}, i.e. \spad{prod(lv_i ** ln_i)}. monomials: % -> List % ++ monomials(p) returns the list of non-zero monomials of polynomial p, i.e. ++ \spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}. univariate : (%,VarSet) -> SparseUnivariatePolynomial(%) ++ univariate(p,v) converts the multivariate polynomial p ++ into a univariate polynomial in v, whose coefficients are still ++ multivariate polynomials (in all the other variables). univariate : % -> SparseUnivariatePolynomial(R) ++ univariate(p) converts the multivariate polynomial p, ++ which should actually involve only one variable, ++ into a univariate polynomial ++ in that variable, whose coefficients are in the ground ring. ++ Error: if polynomial is genuinely multivariate mainVariable : % -> Union(VarSet,"failed") ++ mainVariable(p) returns the biggest variable which actually ++ occurs in the polynomial p, or "failed" if no variables are ++ present. ++ fails precisely if polynomial satisfies ground? minimumDegree : (%,VarSet) -> NonNegativeInteger ++ minimumDegree(p,v) gives the minimum degree of polynomial p ++ with respect to v, i.e. viewed a univariate polynomial in v minimumDegree : (%,List(VarSet)) -> List(NonNegativeInteger) ++ minimumDegree(p, lv) gives the list of minimum degrees of the ++ polynomial p with respect to each of the variables in the list lv monicDivide : (%,%,VarSet) -> Record(quotient:%,remainder:%) ++ monicDivide(a,b,v) divides the polynomial a by the polynomial b, ++ with each viewed as a univariate polynomial in v returning ++ both the quotient and remainder. ++ Error: if b is not monic with respect to v. monomial : (%,VarSet,NonNegativeInteger) -> % ++ monomial(a,x,n) creates the monomial \spad{a*x**n} where \spad{a} is ++ a polynomial, x is a variable and n is a nonnegative integer. monomial : (%,List VarSet,List NonNegativeInteger) -> % ++ monomial(a,[v1..vn],[e1..en]) returns \spad{a*prod(vi**ei)}. multivariate : (SparseUnivariatePolynomial(R),VarSet) -> % ++ multivariate(sup,v) converts an anonymous univariable ++ polynomial sup to a polynomial in the variable v. multivariate : (SparseUnivariatePolynomial(%),VarSet) -> % ++ multivariate(sup,v) converts an anonymous univariable ++ polynomial sup to a polynomial in the variable v. isPlus: % -> Union(List %, "failed") ++ isPlus(p) returns \spad{[m1,...,mn]} if polynomial \spad{p = m1 + ... + mn} and ++ \spad{n >= 2} and each mi is a nonzero monomial. isTimes: % -> Union(List %, "failed") ++ isTimes(p) returns \spad{[a1,...,an]} if polynomial \spad{p = a1 ... an} ++ and \spad{n >= 2}, and, for each i, ai is either a nontrivial constant in R or else of the ++ form \spad{x**e}, where \spad{e > 0} is an integer and x in a member of VarSet. isExpt: % -> Union(Record(var:VarSet, exponent:NonNegativeInteger),_ "failed") ++ isExpt(p) returns \spad{[x, n]} if polynomial p has the form \spad{x**n} and \spad{n > 0}. totalDegree : % -> NonNegativeInteger ++ totalDegree(p) returns the largest sum over all monomials ++ of all exponents of a monomial. totalDegree : (%,List VarSet) -> NonNegativeInteger ++ totalDegree(p, lv) returns the maximum sum (over all monomials of polynomial p) ++ of the variables in the list lv. variables : % -> List(VarSet) ++ variables(p) returns the list of those variables actually ++ appearing in the polynomial p. primitiveMonomials: % -> List % ++ primitiveMonomials(p) gives the list of monomials of the ++ polynomial p with their coefficients removed. ++ Note: \spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}. -- OrderedRing view removed to allow EXPR to define abs --if R has OrderedRing then OrderedRing if (R has ConvertibleTo InputForm) and (VarSet has ConvertibleTo InputForm) then ConvertibleTo InputForm if (R has ConvertibleTo Pattern Integer) and (VarSet has ConvertibleTo Pattern Integer) then ConvertibleTo Pattern Integer if (R has ConvertibleTo Pattern Float) and (VarSet has ConvertibleTo Pattern Float) then ConvertibleTo Pattern Float if (R has PatternMatchable Integer) and (VarSet has PatternMatchable Integer) then PatternMatchable Integer if (R has PatternMatchable Float) and (VarSet has PatternMatchable Float) then PatternMatchable Float if R has CommutativeRing then resultant : (%,%,VarSet) -> % ++ resultant(p,q,v) returns the resultant of the polynomials ++ p and q with respect to the variable v. discriminant : (%,VarSet) -> % ++ discriminant(p,v) returns the disriminant of the polynomial p ++ with respect to the variable v. if R has GcdDomain then GcdDomain content: (%,VarSet) -> % ++ content(p,v) is the gcd of the coefficients of the polynomial p ++ when p is viewed as a univariate polynomial with respect to the ++ variable v. ++ Thus, for polynomial 7*x**2*y + 14*x*y**2, the gcd of the ++ coefficients with respect to x is 7*y. primitivePart: % -> % ++ primitivePart(p) returns the unitCanonical associate of the ++ polynomial p with its content divided out. primitivePart: (%,VarSet) -> % ++ primitivePart(p,v) returns the unitCanonical associate of the ++ polynomial p with its content with respect to the variable v ++ divided out. squareFree: % -> Factored % ++ squareFree(p) returns the square free factorization of the ++ polynomial p. squareFreePart: % -> % ++ squareFreePart(p) returns product of all the irreducible factors ++ of polynomial p each taken with multiplicity one. -- assertions if R has canonicalUnitNormal then canonicalUnitNormal ++ we can choose a unique representative for each ++ associate class. ++ This normalization is chosen to be normalization of ++ leading coefficient (by default). if R has PolynomialFactorizationExplicit then PolynomialFactorizationExplicit add p:% v:VarSet ln:List NonNegativeInteger lv:List VarSet n:NonNegativeInteger pp,qq:SparseUnivariatePolynomial % eval(p:%, l:List Equation %) == empty? l => p for e in l repeat retractIfCan(lhs e)@Union(VarSet,"failed") case "failed" => error "cannot find a variable to evaluate" lvar:=[retract(lhs e)@VarSet for e in l] eval(p, lvar,[rhs e for e in l]$List(%)) monomials p == -- zero? p => empty() -- concat(leadingMonomial p, monomials reductum p) -- replaced by sequential version for efficiency, by WMSIT, 7/30/90 ml:= empty$List(%) while p ~= 0 repeat ml:=concat(leadingMonomial p, ml) p:= reductum p reverse ml isPlus p == empty? rest(l := monomials p) => "failed" l isTimes p == empty?(lv := variables p) or not monomial? p => "failed" l := [monomial(1, v, degree(p, v)) for v in lv] one?(r := leadingCoefficient p) => empty? rest lv => "failed" l concat(r::%, l) isExpt p == (u := mainVariable p) case "failed" => "failed" p = monomial(1, u::VarSet, d := degree(p, u::VarSet)) => [u::VarSet, d] "failed" coefficient(p,v,n) == coefficient(univariate(p,v),n) coefficient(p,lv,ln) == empty? lv => empty? ln => p error "mismatched lists in coefficient" empty? ln => error "mismatched lists in coefficient" coefficient(coefficient(univariate(p,first lv),first ln), rest lv,rest ln) monomial(p,lv,ln) == empty? lv => empty? ln => p error "mismatched lists in monomial" empty? ln => error "mismatched lists in monomial" monomial(monomial(p,first lv, first ln),rest lv, rest ln) retract(p:%):VarSet == q := mainVariable(p)::VarSet q::% = p => q error "Polynomial is not a single variable" retractIfCan(p:%):Union(VarSet, "failed") == ((q := mainVariable p) case VarSet) and (q::VarSet::% = p) => q "failed" mkPrim(p:%):% == monomial(1,degree p) primitiveMonomials p == [mkPrim q for q in monomials p] totalDegree p == ground? p => 0 u := univariate(p, mainVariable(p)::VarSet) d: NonNegativeInteger := 0 while u ~= 0 repeat d := max(d, degree u + totalDegree leadingCoefficient u) u := reductum u d totalDegree(p,lv) == ground? p => 0 u := univariate(p, v:=(mainVariable(p)::VarSet)) d: NonNegativeInteger := 0 w: NonNegativeInteger := 0 if member?(v, lv) then w:=1 while u ~= 0 repeat d := max(d, w*(degree u) + totalDegree(leadingCoefficient u,lv)) u := reductum u d if R has CommutativeRing then resultant(p1,p2,mvar) == resultant(univariate(p1,mvar),univariate(p2,mvar)) discriminant(p,var) == discriminant(univariate(p,var)) if R has IntegralDomain then allMonoms(l:List %):List(%) == removeDuplicates_! concat [primitiveMonomials p for p in l] P2R(p:%, b:List E, n:NonNegativeInteger):Vector(R) == w := new(n, 0)$Vector(R) for i in minIndex w .. maxIndex w for bj in b repeat qsetelt_!(w, i, coefficient(p, bj)) w eq2R(l:List %, b:List E):Matrix(R) == matrix [[coefficient(p, bj) for p in l] for bj in b] reducedSystem(m:Matrix %):Matrix(R) == l := listOfLists m b := removeDuplicates_! concat [allMonoms r for r in l]$List(List(%)) d := [degree bj for bj in b] mm := eq2R(first l, d) l := rest l while not empty? l repeat mm := vertConcat(mm, eq2R(first l, d)) l := rest l mm reducedSystem(m:Matrix %, v:Vector %): Record(mat:Matrix R, vec:Vector R) == l := listOfLists m r := entries v b : List % := removeDuplicates_! concat(allMonoms r, concat [allMonoms s for s in l]$List(List(%))) d := [degree bj for bj in b] n := #d mm := eq2R(first l, d) w := P2R(first r, d, n) l := rest l r := rest r while not empty? l repeat mm := vertConcat(mm, eq2R(first l, d)) w := concat(w, P2R(first r, d, n)) l := rest l r := rest r [mm, w] if R has PolynomialFactorizationExplicit then -- we might be in trouble if its actually only -- a univariate polynomial category - have to remember to -- over-ride these in UnivariatePolynomialCategory PFBR ==>PolynomialFactorizationByRecursion(R,E,VarSet,%) gcdPolynomial(pp,qq) == gcdPolynomial(pp,qq)$GeneralPolynomialGcdPackage(E,VarSet,R,%) solveLinearPolynomialEquation(lpp,pp) == solveLinearPolynomialEquationByRecursion(lpp,pp)$PFBR factorPolynomial(pp) == factorByRecursion(pp)$PFBR factorSquareFreePolynomial(pp) == factorSquareFreeByRecursion(pp)$PFBR factor p == v:Union(VarSet,"failed"):=mainVariable p v case "failed" => ansR:=factor leadingCoefficient p makeFR(unit(ansR)::%, [[w.flg,w.fctr::%,w.xpnt] for w in factorList ansR]) up:SparseUnivariatePolynomial %:=univariate(p,v) ansSUP:=factorByRecursion(up)$PFBR makeFR(multivariate(unit(ansSUP),v), [[ww.flg,multivariate(ww.fctr,v),ww.xpnt] for ww in factorList ansSUP]) if R has CharacteristicNonZero then mat: Matrix % conditionP mat == ll:=listOfLists transpose mat -- hence each list corresponds to a -- column, i.e. to one variable llR:List List R := [ empty() for z in first ll] monslist:List List % := empty() ch:=characteristic()$% for l in ll repeat mons:= "setUnion"/[primitiveMonomials u for u in l] redmons:List % :=[] for m in mons repeat vars:=variables m degs:=degree(m,vars) deg1:List NonNegativeInteger deg1:=[ ((nd:=d:Integer exquo ch:Integer) case "failed" => return "failed" ; nd::Integer::NonNegativeInteger) for d in degs ] redmons:=[monomial(1,vars,deg1),:redmons] llR:=[[ground coefficient(u,vars,degs),:v] for u in l for v in llR] monslist:=[redmons,:monslist] ans:=conditionP transpose matrix llR ans case "failed" => "failed" i:NonNegativeInteger:=0 [ +/[m*(ans.(i:=i+1))::% for m in mons ] for mons in monslist] if R has CharacteristicNonZero then charthRootlv: (%,List VarSet,NonNegativeInteger) -> Union(%,"failed") charthRoot p == vars:= variables p empty? vars => ans := charthRoot ground p ans case "failed" => "failed" ans::R::% ch:=characteristic()$% charthRootlv(p,vars,ch) charthRootlv(p,vars,ch) == empty? vars => ans := charthRoot ground p ans case "failed" => "failed" ans::R::% v:=first vars vars:=rest vars d:=degree(p,v) ans:% := 0 while (d>0) repeat (dd:=(d::Integer exquo ch::Integer)) case "failed" => return "failed" cp:=coefficient(p,v,d) p:=p-monomial(cp,v,d) ansx:=charthRootlv(cp,vars,ch) ansx case "failed" => return "failed" d:=degree(p,v) ans:=ans+monomial(ansx,v,dd::Integer::NonNegativeInteger) ansx:=charthRootlv(p,vars,ch) ansx case "failed" => return "failed" return ans+ansx monicDivide(p1,p2,mvar) == result:=monicDivide(univariate(p1,mvar),univariate(p2,mvar)) [multivariate(result.quotient,mvar), multivariate(result.remainder,mvar)] if R has GcdDomain then if R has EuclideanDomain and R has CharacteristicZero then squareFree p == squareFree(p)$MultivariateSquareFree(E,VarSet,R,%) else squareFree p == squareFree(p)$PolynomialSquareFree(VarSet,E,R,%) squareFreePart p == unit(s := squareFree p) * */[f.factor for f in factors s] content(p,v) == content univariate(p,v) primitivePart p == unitNormal((p exquo content p) ::%).canonical primitivePart(p,v) == unitNormal((p exquo content(p,v)) ::%).canonical before?(p:%, q:%) == (dp:= degree p) < (dq := degree q) => before?(0, leadingCoefficient q) dq < dp => before?(leadingCoefficient p,0) before?(leadingCoefficient(p - q),0) if (R has PatternMatchable Integer) and (VarSet has PatternMatchable Integer) then patternMatch(p:%, pat:Pattern Integer, l:PatternMatchResult(Integer, %)) == patternMatch(p, pat, l)$PatternMatchPolynomialCategory(Integer,E,VarSet,R,%) if (R has PatternMatchable Float) and (VarSet has PatternMatchable Float) then patternMatch(p:%, pat:Pattern Float, l:PatternMatchResult(Float, %)) == patternMatch(p, pat, l)$PatternMatchPolynomialCategory(Float,E,VarSet,R,%) if (R has ConvertibleTo Pattern Integer) and (VarSet has ConvertibleTo Pattern Integer) then convert(x:%):Pattern(Integer) == map(convert, convert, x)$PolynomialCategoryLifting(E,VarSet,R,%,Pattern Integer) if (R has ConvertibleTo Pattern Float) and (VarSet has ConvertibleTo Pattern Float) then convert(x:%):Pattern(Float) == map(convert, convert, x)$PolynomialCategoryLifting(E, VarSet, R, %, Pattern Float) if (R has ConvertibleTo InputForm) and (VarSet has ConvertibleTo InputForm) then convert(p:%):InputForm == map(convert, convert, p)$PolynomialCategoryLifting(E,VarSet,R,%,InputForm) @ \section{package POLYLIFT PolynomialCategoryLifting} <>= )abbrev package POLYLIFT PolynomialCategoryLifting ++ Author: Manuel Bronstein ++ Date Created: ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ This package provides a very general map function, which ++ given a set S and polynomials over R with maps from the ++ variables into S and the coefficients into S, maps polynomials ++ into S. S is assumed to support \spad{+}, \spad{*} and \spad{**}. PolynomialCategoryLifting(E,Vars,R,P,S): Exports == Implementation where E : OrderedAbelianMonoidSup Vars: OrderedSet R : Ring P : PolynomialCategory(R, E, Vars) S : SetCategory with "+" : (%, %) -> % "*" : (%, %) -> % "**": (%, NonNegativeInteger) -> % Exports ==> with map: (Vars -> S, R -> S, P) -> S ++ map(varmap, coefmap, p) takes a ++ varmap, a mapping from the variables of polynomial p into S, ++ coefmap, a mapping from coefficients of p into S, and p, and ++ produces a member of S using the corresponding arithmetic. ++ in S Implementation ==> add map(fv, fc, p) == (x1 := mainVariable p) case "failed" => fc leadingCoefficient p up := univariate(p, x1::Vars) t := fv(x1::Vars) ans:= fc 0 while not ground? up repeat ans := ans + map(fv,fc, leadingCoefficient up) * t ** (degree up) up := reductum up ans + map(fv, fc, leadingCoefficient up) @ \section{category UPOLYC UnivariatePolynomialCategory} <>= )abbrev category UPOLYC UnivariatePolynomialCategory ++ Author: ++ Date Created: ++ Date Last Updated: ++ Basic Functions: Ring, monomial, coefficient, reductum, differentiate, ++ elt, map, resultant, discriminant ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ The category of univariate polynomials over a ring R. ++ No particular model is assumed - implementations can be either ++ sparse or dense. UnivariatePolynomialCategory(R:Ring): Category == Join(PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet), Eltable(R, R), Eltable(%, %), DifferentialRing, DifferentialExtension R) with vectorise : (%,NonNegativeInteger) -> Vector R ++ vectorise(p, n) returns \spad{[a0,...,a(n-1)]} where ++ \spad{p = a0 + a1*x + ... + a(n-1)*x**(n-1)} + higher order terms. ++ The degree of polynomial p can be different from \spad{n-1}. makeSUP: % -> SparseUnivariatePolynomial R ++ makeSUP(p) converts the polynomial p to be of type ++ SparseUnivariatePolynomial over the same coefficients. unmakeSUP: SparseUnivariatePolynomial R -> % ++ unmakeSUP(sup) converts sup of type \spadtype{SparseUnivariatePolynomial(R)} ++ to be a member of the given type. ++ Note: converse of makeSUP. multiplyExponents: (%,NonNegativeInteger) -> % ++ multiplyExponents(p,n) returns a new polynomial resulting from ++ multiplying all exponents of the polynomial p by the non negative ++ integer n. divideExponents: (%,NonNegativeInteger) -> Union(%,"failed") ++ divideExponents(p,n) returns a new polynomial resulting from ++ dividing all exponents of the polynomial p by the non negative ++ integer n, or "failed" if some exponent is not exactly divisible ++ by n. monicDivide: (%,%) -> Record(quotient:%,remainder:%) ++ monicDivide(p,q) divide the polynomial p by the monic polynomial q, ++ returning the pair \spad{[quotient, remainder]}. ++ Error: if q isn't monic. -- These three are for Karatsuba karatsubaDivide: (%,NonNegativeInteger) -> Record(quotient:%,remainder:%) ++ \spad{karatsubaDivide(p,n)} returns the same as \spad{monicDivide(p,monomial(1,n))} shiftRight: (%,NonNegativeInteger) -> % ++ \spad{shiftRight(p,n)} returns \spad{monicDivide(p,monomial(1,n)).quotient} shiftLeft: (%,NonNegativeInteger) -> % ++ \spad{shiftLeft(p,n)} returns \spad{p * monomial(1,n)} pseudoRemainder: (%,%) -> % ++ pseudoRemainder(p,q) = r, for polynomials p and q, returns the remainder when ++ \spad{p' := p*lc(q)**(deg p - deg q + 1)} ++ is pseudo right-divided by q, i.e. \spad{p' = s q + r}. differentiate: (%, R -> R, %) -> % ++ differentiate(p, d, x') extends the R-derivation d to an ++ extension D in \spad{R[x]} where Dx is given by x', and returns \spad{Dp}. if R has StepThrough then StepThrough if R has CommutativeRing then discriminant: % -> R ++ discriminant(p) returns the discriminant of the polynomial p. resultant: (%,%) -> R ++ resultant(p,q) returns the resultant of the polynomials p and q. if R has IntegralDomain then Eltable(Fraction %, Fraction %) elt : (Fraction %, Fraction %) -> Fraction % ++ elt(a,b) evaluates the fraction of univariate polynomials \spad{a} ++ with the distinguished variable replaced by b. order: (%, %) -> NonNegativeInteger ++ order(p, q) returns the largest n such that \spad{q**n} divides polynomial p ++ i.e. the order of \spad{p(x)} at \spad{q(x)=0}. subResultantGcd: (%,%) -> % ++ subResultantGcd(p,q) computes the gcd of the polynomials p and q ++ using the SubResultant GCD algorithm. composite: (%, %) -> Union(%, "failed") ++ composite(p, q) returns h if \spad{p = h(q)}, and "failed" no such h exists. composite: (Fraction %, %) -> Union(Fraction %, "failed") ++ composite(f, q) returns h if f = h(q), and "failed" is no such h exists. pseudoQuotient: (%,%) -> % ++ pseudoQuotient(p,q) returns r, the quotient when ++ \spad{p' := p*lc(q)**(deg p - deg q + 1)} ++ is pseudo right-divided by q, i.e. \spad{p' = s q + r}. pseudoDivide: (%, %) -> Record(coef:R, quotient: %, remainder:%) ++ pseudoDivide(p,q) returns \spad{[c, q, r]}, when ++ \spad{p' := p*lc(q)**(deg p - deg q + 1) = c * p} ++ is pseudo right-divided by q, i.e. \spad{p' = s q + r}. if R has GcdDomain then separate: (%, %) -> Record(primePart:%, commonPart: %) ++ separate(p, q) returns \spad{[a, b]} such that polynomial \spad{p = a b} and ++ \spad{a} is relatively prime to q. if R has Field then EuclideanDomain additiveValuation ++ euclideanSize(a*b) = euclideanSize(a) + euclideanSize(b) elt : (Fraction %, R) -> R ++ elt(a,r) evaluates the fraction of univariate polynomials \spad{a} ++ with the distinguished variable replaced by the constant r. if R has Algebra Fraction Integer then integrate: % -> % ++ integrate(p) integrates the univariate polynomial p with respect ++ to its distinguished variable. add pp,qq: SparseUnivariatePolynomial % variables(p) == zero? p or zero?(degree p) => [] [create()] degree(p:%,v:SingletonAsOrderedSet) == degree p totalDegree(p:%,lv:List SingletonAsOrderedSet) == empty? lv => 0 totalDegree p degree(p:%,lv:List SingletonAsOrderedSet) == empty? lv => [] [degree p] eval(p:%,lv: List SingletonAsOrderedSet,lq: List %):% == empty? lv => p not empty? rest lv => error "can only eval a univariate polynomial once" eval(p,first lv,first lq)$% eval(p:%,v:SingletonAsOrderedSet,q:%):% == p(q) eval(p:%,lv: List SingletonAsOrderedSet,lr: List R):% == empty? lv => p not empty? rest lv => error "can only eval a univariate polynomial once" eval(p,first lv,first lr)$% eval(p:%,v:SingletonAsOrderedSet,r:R):% == p(r)::% eval(p:%,le:List Equation %):% == empty? le => p not empty? rest le => error "can only eval a univariate polynomial once" mainVariable(lhs first le) case "failed" => p p(rhs first le) mainVariable(p:%) == zero? degree p => "failed" create()$SingletonAsOrderedSet minimumDegree(p:%,v:SingletonAsOrderedSet) == minimumDegree p minimumDegree(p:%,lv:List SingletonAsOrderedSet) == empty? lv => [] [minimumDegree p] monomial(p:%,v:SingletonAsOrderedSet,n:NonNegativeInteger) == mapExponents(#1+n,p) coerce(v:SingletonAsOrderedSet):% == monomial(1,1) makeSUP p == zero? p => 0 monomial(leadingCoefficient p,degree p) + makeSUP reductum p unmakeSUP sp == zero? sp => 0 monomial(leadingCoefficient sp,degree sp) + unmakeSUP reductum sp karatsubaDivide(p:%,n:NonNegativeInteger) == monicDivide(p,monomial(1,n)) shiftRight(p:%,n:NonNegativeInteger) == monicDivide(p,monomial(1,n)).quotient shiftLeft(p:%,n:NonNegativeInteger) == p * monomial(1,n) if R has PolynomialFactorizationExplicit then PFBRU ==>PolynomialFactorizationByRecursionUnivariate(R,%) pp,qq:SparseUnivariatePolynomial % lpp:List SparseUnivariatePolynomial % SupR ==> SparseUnivariatePolynomial R sp:SupR solveLinearPolynomialEquation(lpp,pp) == solveLinearPolynomialEquationByRecursion(lpp,pp)$PFBRU factorPolynomial(pp) == factorByRecursion(pp)$PFBRU factorSquareFreePolynomial(pp) == factorSquareFreeByRecursion(pp)$PFBRU import FactoredFunctions2(SupR,S) factor p == zero? degree p => ansR:=factor leadingCoefficient p makeFR(unit(ansR)::%, [[w.flg,w.fctr::%,w.xpnt] for w in factorList ansR]) map(unmakeSUP,factorPolynomial(makeSUP p)$R) vectorise(p, n) == m := minIndex(v := new(n, 0)$Vector(R)) for i in minIndex v .. maxIndex v repeat qsetelt_!(v, i, coefficient(p, (i - m)::NonNegativeInteger)) v retract(p:%):R == zero? p => 0 zero? degree p => leadingCoefficient p error "Polynomial is not of degree 0" retractIfCan(p:%):Union(R, "failed") == zero? p => 0 zero? degree p => leadingCoefficient p "failed" if R has StepThrough then init() == init()$R::% nextItemInner: % -> Union(%,"failed") nextItemInner(n) == zero? n => nextItem(0$R)::R::% -- assumed not to fail zero? degree n => nn:=nextItem leadingCoefficient n nn case "failed" => "failed" nn::R::% n1:=reductum n n2:=nextItemInner n1 -- try stepping the reductum n2 case % => monomial(leadingCoefficient n,degree n) + n2 1+degree n1 < degree n => -- there was a hole between lt n and n1 monomial(leadingCoefficient n,degree n)+ monomial(nextItem(init()$R)::R,1+degree n1) n3:=nextItem leadingCoefficient n n3 case "failed" => "failed" monomial(n3,degree n) nextItem(n) == n1:=nextItemInner n n1 case "failed" => monomial(nextItem(init()$R)::R,1+degree(n)) n1 if R has GcdDomain then content(p:%,v:SingletonAsOrderedSet) == content(p)::% primeFactor: (%, %) -> % primeFactor(p, q) == (p1 := (p exquo gcd(p, q))::%) = p => p primeFactor(p1, q) separate(p, q) == a := primeFactor(p, q) [a, (p exquo a)::%] if R has CommutativeRing then differentiate(x:%, deriv:R -> R, x':%) == d:% := 0 while (dg := degree x) > 0 repeat lc := leadingCoefficient x d := d + x' * monomial(dg * lc, (dg - 1)::NonNegativeInteger) + monomial(deriv lc, dg) x := reductum x d + deriv(leadingCoefficient x)::% else ncdiff: (NonNegativeInteger, %) -> % -- computes d(x**n) given dx = x', non-commutative case ncdiff(n, x') == zero? n => 0 zero?(n1 := (n - 1)::NonNegativeInteger) => x' x' * monomial(1, n1) + monomial(1, 1) * ncdiff(n1, x') differentiate(x:%, deriv:R -> R, x':%) == d:% := 0 while (dg := degree x) > 0 repeat lc := leadingCoefficient x d := d + monomial(deriv lc, dg) + lc * ncdiff(dg, x') x := reductum x d + deriv(leadingCoefficient x)::% differentiate(x:%, deriv:R -> R) == differentiate(x, deriv, 1$%)$% differentiate(x:%) == d:% := 0 while (dg := degree x) > 0 repeat d := d + monomial(dg * leadingCoefficient x, (dg - 1)::NonNegativeInteger) x := reductum x d differentiate(x:%,v:SingletonAsOrderedSet) == differentiate x if R has IntegralDomain then elt(g:Fraction %, f:Fraction %) == ((numer g) f) / ((denom g) f) pseudoQuotient(p, q) == (n := degree(p)::Integer - degree q + 1) < 1 => 0 ((leadingCoefficient(q)**(n::NonNegativeInteger) * p - pseudoRemainder(p, q)) exquo q)::% pseudoDivide(p, q) == (n := degree(p)::Integer - degree q + 1) < 1 => [1, 0, p] prem := pseudoRemainder(p, q) lc := leadingCoefficient(q)**(n::NonNegativeInteger) [lc,((lc*p - prem) exquo q)::%, prem] composite(f:Fraction %, q:%) == (n := composite(numer f, q)) case "failed" => "failed" (d := composite(denom f, q)) case "failed" => "failed" n::% / d::% composite(p:%, q:%) == ground? p => p cqr := pseudoDivide(p, q) ground?(cqr.remainder) and ((v := cqr.remainder exquo cqr.coef) case %) and ((u := composite(cqr.quotient, q)) case %) and ((w := (u::%) exquo cqr.coef) case %) => v::% + monomial(1, 1) * w::% "failed" elt(p:%, f:Fraction %) == zero? p => 0 ans:Fraction(%) := (leadingCoefficient p)::%::Fraction(%) n := degree p while not zero?(p:=reductum p) repeat ans := ans * f ** (n - (n := degree p))::NonNegativeInteger + (leadingCoefficient p)::%::Fraction(%) zero? n => ans ans * f ** n order(p, q) == zero? p => error "order: arguments must be nonzero" degree(q) < 1 => error "order: place must be non-trivial" ans:NonNegativeInteger := 0 repeat (u := p exquo q) case "failed" => return ans p := u::% ans := ans + 1 if R has GcdDomain then squareFree(p:%) == squareFree(p)$UnivariatePolynomialSquareFree(R, %) squareFreePart(p:%) == squareFreePart(p)$UnivariatePolynomialSquareFree(R, %) if R has PolynomialFactorizationExplicit then gcdPolynomial(pp,qq) == zero? pp => unitCanonical qq -- subResultantGcd can't handle 0 zero? qq => unitCanonical pp unitCanonical(gcd(content (pp),content(qq))* primitivePart subResultantGcd(primitivePart pp,primitivePart qq)) squareFreePolynomial pp == squareFree(pp)$UnivariatePolynomialSquareFree(%, SparseUnivariatePolynomial %) if R has Field then elt(f:Fraction %, r:R) == ((numer f) r) / ((denom f) r) euclideanSize x == zero? x => error "euclideanSize called on 0 in Univariate Polynomial" degree x divide(x,y) == zero? y => error "division by 0 in Univariate Polynomials" quot:=0 lc := inv leadingCoefficient y while not zero?(x) and (degree x >= degree y) repeat f:=lc*leadingCoefficient x n:=(degree x - degree y)::NonNegativeInteger quot:=quot+monomial(f,n) x:=x-monomial(f,n)*y [quot,x] if R has Algebra Fraction Integer then integrate p == ans:% := 0 while p ~= 0 repeat l := leadingCoefficient p d := 1 + degree p ans := ans + inv(d::Fraction(Integer)) * monomial(l, d) p := reductum p ans @ \section{package UPOLYC2 UnivariatePolynomialCategoryFunctions2} <>= )abbrev package UPOLYC2 UnivariatePolynomialCategoryFunctions2 ++ Author: ++ Date Created: ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ Mapping from polynomials over R to polynomials over S ++ given a map from R to S assumed to send zero to zero. UnivariatePolynomialCategoryFunctions2(R,PR,S,PS): Exports == Impl where R, S: Ring PR : UnivariatePolynomialCategory R PS : UnivariatePolynomialCategory S Exports ==> with map: (R -> S, PR) -> PS ++ map(f, p) takes a function f from R to S, ++ and applies it to each (non-zero) coefficient of a polynomial p ++ over R, getting a new polynomial over S. ++ Note: since the map is not applied to zero elements, it may map zero ++ to zero. Impl ==> add map(f, p) == ans:PS := 0 while p ~= 0 repeat ans := ans + monomial(f leadingCoefficient p, degree p) p := reductum p ans @ \section{package COMMUPC CommuteUnivariatePolynomialCategory} <>= )abbrev package COMMUPC CommuteUnivariatePolynomialCategory ++ Author: Manuel Bronstein ++ Date Created: ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A package for swapping the order of two variables in a tower of two ++ UnivariatePolynomialCategory extensions. CommuteUnivariatePolynomialCategory(R, UP, UPUP): Exports == Impl where R : Ring UP : UnivariatePolynomialCategory R UPUP: UnivariatePolynomialCategory UP N ==> NonNegativeInteger Exports ==> with swap: UPUP -> UPUP ++ swap(p(x,y)) returns p(y,x). Impl ==> add makePoly: (UP, N) -> UPUP -- converts P(x,y) to P(y,x) swap poly == ans:UPUP := 0 while poly ~= 0 repeat ans := ans + makePoly(leadingCoefficient poly, degree poly) poly := reductum poly ans makePoly(poly, d) == ans:UPUP := 0 while poly ~= 0 repeat ans := ans + monomial(monomial(leadingCoefficient poly, d), degree poly) poly := reductum poly ans @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. @ <<*>>= <> <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}