\documentclass{article} \usepackage{open-axiom} \begin{document} \title{\$SPAD/src/algebra unifact.spad} \author{Patrizia Gianni} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{package UNIFACT UnivariateFactorize} <>= )abbrev package UNIFACT UnivariateFactorize ++ Factorisation of univariate polynomials with integer coefficients ++ Author: Patrizia Gianni ++ Date Created: ??? ++ Date Last Updated: December 1993 ++ Description: ++ Package for the factorization of univariate polynomials with integer ++ coefficients. The factorization is done by "lifting" (HENSEL) the ++ factorization over a finite field. UnivariateFactorize(ZP) : public == private where Z ==> Integer PI ==> PositiveInteger NNI ==> NonNegativeInteger SUPZ ==> SparseUnivariatePolynomial Z ZP : UnivariatePolynomialCategory Z FR ==> Factored ZP fUnion ==> Union("nil", "sqfr", "irred", "prime") FFE ==> Record(flg:fUnion, fctr:ZP, xpnt:Z) ParFact ==> Record(irr: ZP,pow: Z) FinalFact ==> Record(contp: Z,factors:List(ParFact)) public == with factor : ZP -> FR ++ factor(m) returns the factorization of m factorSquareFree : ZP -> FR ++ factorSquareFree(m) returns the factorization of m square free ++ polynomial henselFact : (ZP,Boolean) -> FinalFact ++ henselFact(m,flag) returns the factorization of m, ++ FinalFact is a Record s.t. FinalFact.contp=content m, ++ FinalFact.factors=List of irreducible factors ++ of m with exponent , if flag =true the polynomial is ++ assumed square free. private == add --- local functions --- henselfact : ZP -> List(ZP) quadratic : ZP -> List(ZP) remp : (Z, PI) -> Z negShiftz : (Z, PI) -> Z negShiftp : (ZP,PI) -> ZP bound : ZP -> PI choose : ZP -> FirstStep eisenstein : ZP -> Boolean isPowerOf2 : Z -> Boolean subMinusX : SUPZ -> ZP sqroot : Z -> Z --- declarations --- CYC ==> CyclotomicPolynomialPackage() DDRecord ==> Record(factor: ZP,degree: Z) DDList ==> List DDRecord FirstStep ==> Record(prime:PI,factors:DDList) ContPrim ==> Record(cont: Z,prim: ZP) import GeneralHenselPackage(Z,ZP) import ModularDistinctDegreeFactorizer ZP factor(m: ZP) == flist := henselFact(m,false) ctp:=unitNormal flist.contp makeFR((ctp.unit)::ZP,cons(["nil",ctp.canonical::ZP,1$Z]$FFE, [["prime",u.irr,u.pow]$FFE for u in flist.factors])) factorSquareFree(m: ZP) == flist := henselFact(m,true) ctp:=unitNormal flist.contp makeFR((ctp.unit)::ZP,cons(["nil",ctp.canonical::ZP,1$Z]$FFE, [["prime",u.irr,u.pow]$FFE for u in flist.factors])) -- Integer square root: returns 0 if t is non-positive sqroot(t: Z): Z == t <= 0 => 0 s:Integer:=t::Integer s:=approxSqrt(s)$IntegerRoots(Integer) t:=s::Z t -- Eisenstein criterion: returns true if polynomial is -- irreducible. Result of false in inconclusive. eisenstein(m : ZP): Boolean == -- calculate the content of the terms after the first c := content reductum m c = 0 => false c = 1 => false -- factor the content -- if there is a prime in the factorization that does not divide -- the leading term and appears to multiplicity 1, and the square -- of this does not divide the last coef, return true. -- Otherwise reurn false. lead := leadingCoefficient m trail := lead m := reductum m while m ~= 0 repeat trail := leadingCoefficient m m:= reductum m fc := factor(c) :: Factored(Z) for r in factors fc repeat if (r.exponent = 1) and (0 ~= (lead rem r.factor)) and (0 ~= (trail rem (r.factor ** 2))) then return true false negShiftz(n: Z,Modulus:PI): Z == if negative? n then n := n+Modulus n > (Modulus quo 2) => n-Modulus n negShiftp(pp: ZP,Modulus:PI): ZP == map(negShiftz(#1,Modulus),pp) -- Choose the bound for the coefficients of factors bound(m: ZP):PI == nm,nmq2,lcm,bin0,bin1:NNI cbound,j : PI k:NNI lcm := abs(leadingCoefficient m)::NNI nm := (degree m)::NNI nmq2:NNI := nm quo 2 norm: Z := sqroot(+/[coefficient(m,k)**2 for k in 0..nm]) if not one? nmq2 then nm := (nmq2-1):NNI else nm := nmq2 bin0 := nm cbound := (bin0*norm+lcm)::PI for i in 2..(nm-1)::NNI repeat bin1 := bin0 bin0 := (bin0*(nm+1-i):NNI) quo i j := (bin0*norm+bin1*lcm)::PI if cbound t+q ;t) numFactors(ddlist:DDList): Z == ans: Z := 0 for dd in ddlist repeat (d := degree(dd.factor)) = 0 => nil ans := ans + ((d pretend Z) exquo dd.degree):: Z ans -- select the prime,try up to 4 primes, -- choose the one yielding the fewest factors, but stopping if -- fewer than 9 factors choose(m: ZP):FirstStep == qSave:PI := 1 ddSave:DDList := [] numberOfFactors: Z := 0 lcm := leadingCoefficient m k: Z := 1 ddRep := 5 disc:ZP:=0 q:PI:=2 while k "next prime" disc:=gcd(m,differentiate m,q) not zero?(degree disc) => "next prime" k := k+1 newdd := ddFact(m,q) ((n := numFactors(newdd)) < 9) => ddSave := newdd qSave := q k := 5 (numberOfFactors = 0) or (n < numberOfFactors) => ddSave := newdd qSave := q numberOfFactors := n [qSave,ddSave]$FirstStep -- Find the factors of m,primitive, square-free, with lc positive -- and mindeg m = 0 henselfact1(m: ZP):List(ZP) == zero? degree m => one? m => [] [m] selected := choose(m) (numFactors(selected.factors) = 1$Z) => [m] q := selected.prime fl := separateFactors(selected.factors,q) --choose the bound cbound := bound(m) completeHensel(m,fl,q,cbound) -- check for possible degree reduction -- could use polynomial decomposition ? henselfact(m: ZP):List ZP == deggcd:=degree m mm:= m while not zero? mm repeat (deggcd:=gcd(deggcd, degree mm); mm:=reductum mm) deggcd>1 and deggcd faclist := henselfact1(divideExponents(m, deggcd)::ZP) "append"/[henselfact1 multiplyExponents(mm, deggcd) for mm in faclist] henselfact1 m quadratic(m: ZP):List(ZP) == d,d2: Z d := coefficient(m,1)**2-4*coefficient(m,0)*coefficient(m,2) d2 := sqroot(d) not zero?(d-d2**2) => [m] alpha: Z := coefficient(m,1)+d2 beta: Z := 2*coefficient(m,2) d := gcd(alpha,beta) if not one? d then alpha := alpha quo d beta := beta quo d m0: ZP := monomial(beta,1)+monomial(alpha,0) cons(m0,[(m exquo m0):: ZP]) isPowerOf2(n : Z): Boolean == n = 1 => true qr : Record(quotient: Z, remainder: Z) := divide(n,2) qr.remainder = 1 => false isPowerOf2 qr.quotient subMinusX(supPol : SUPZ): ZP == minusX : SUPZ := monomial(-1,1)$SUPZ (elt(supPol,minusX)$SUPZ) : ZP -- Factorize the polynomial m, test=true if m is known to be -- square-free, false otherwise. -- FinalFact.contp=content m, FinalFact.factors=List of irreducible -- factors with exponent . henselFact(m: ZP,test:Boolean):FinalFact == factorlist : List(ParFact) := [] c : Z -- make m primitive c := content m m := (m exquo c)::ZP -- make the lc m positive if negative? leadingCoefficient m then c := -c m := -m -- is x**d factor of m? if positive?(d := minimumDegree m) then m := (monicDivide(m,monomial(1,d))).quotient factorlist := [[monomial(1,1),d]$ParFact] d := degree m -- is m constant? d=0 => [c,factorlist]$FinalFact -- is m linear? d=1 => [c,cons([m,1]$ParFact,factorlist)]$FinalFact -- does m satisfy Eisenstein's criterion? eisenstein m => [c,cons([m,1]$ParFact,factorlist)]$FinalFact lcPol : ZP := leadingCoefficient(m) :: ZP -- is m cyclotomic (x**n - 1)? -lcPol = reductum(m) => -- if true, both will = 1 for fac in (cyclotomicDecomposition(degree m)$CYC : List ZP) repeat factorlist := cons([fac,1]$ParFact,factorlist) [c,factorlist]$FinalFact -- is m odd cyclotomic (x**(2*n+1) + 1)? odd?(d) and (lcPol = reductum(m)) => for sfac in cyclotomicDecomposition(degree m)$CYC repeat fac:=subMinusX sfac if negative? leadingCoefficient fac then fac := -fac factorlist := cons([fac,1]$ParFact,factorlist) [c,factorlist]$FinalFact -- is the poly of the form x**n + 1 with n a power of 2? -- if so, then irreducible isPowerOf2(d) and (lcPol = reductum(m)) => factorlist := cons([m,1]$ParFact,factorlist) [c,factorlist]$FinalFact -- is m quadratic? d=2 => lfq:List(ZP) := quadratic m #lfq=1 => [c,cons([lfq.first,1]$ParFact,factorlist)]$FinalFact (lf0,lf1) := (lfq.first,second lfq) if lf0=lf1 then factorlist := cons([lf0,2]$ParFact,factorlist) else factorlist := append([[v,1]$ParFact for v in lfq],factorlist) [c,factorlist]$FinalFact -- m is square-free test => fln := henselfact(m) [c,append(factorlist,[[pf,1]$ParFact for pf in fln])]$FinalFact -- find the square-free decomposition of m irrFact := squareFree(m) llf := factors irrFact -- factorize the square-free primitive terms for l1 in llf repeat d1 := l1.exponent pol := l1.factor degree pol=1 => factorlist := cons([pol,d1]$ParFact,factorlist) degree pol=2 => fln := quadratic(pol) factorlist := append([[pf,d1]$ParFact for pf in fln],factorlist) fln := henselfact(pol) factorlist := append([[pf,d1]$ParFact for pf in fln],factorlist) [c,factorlist]$FinalFact @ \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}