/usr/share/axiom-20170501/src/algebra/PFBRU.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 | )abbrev package PFBRU PolynomialFactorizationByRecursionUnivariate
++ Description:
++ PolynomialFactorizationByRecursionUnivariate
++ R is a \spadfun{PolynomialFactorizationExplicit} domain,
++ S is univariate polynomials over R
++ We are interested in handling SparseUnivariatePolynomials over
++ S, is a variable we shall call z
PolynomialFactorizationByRecursionUnivariate(R, S) : SIG == CODE where
R : PolynomialFactorizationExplicit
S : UnivariatePolynomialCategory(R)
PI ==> PositiveInteger
SupR ==> SparseUnivariatePolynomial R
SupSupR ==> SparseUnivariatePolynomial SupR
SupS ==> SparseUnivariatePolynomial S
SupSupS ==> SparseUnivariatePolynomial SupS
LPEBFS ==> LinearPolynomialEquationByFractions(S)
SIG ==> with
solveLinearPolynomialEquationByRecursion: (List SupS, SupS) ->
Union(List SupS,"failed")
++ \spad{solveLinearPolynomialEquationByRecursion([p1,...,pn],p)}
++ returns the list of polynomials \spad{[q1,...,qn]}
++ such that \spad{sum qi/pi = p / prod pi}, a
++ recursion step for solveLinearPolynomialEquation
++ as defined in \spadfun{PolynomialFactorizationExplicit} category
++ (see \spadfun{solveLinearPolynomialEquation}).
++ If no such list of qi exists, then "failed" is returned.
factorByRecursion: SupS -> Factored SupS
++ factorByRecursion(p) factors polynomial p. This function
++ performs the recursion step for factorPolynomial,
++ as defined in \spadfun{PolynomialFactorizationExplicit} category
++ (see \spadfun{factorPolynomial})
factorSquareFreeByRecursion: SupS -> Factored SupS
++ factorSquareFreeByRecursion(p) returns the square free
++ factorization of p. This functions performs
++ the recursion step for factorSquareFreePolynomial,
++ as defined in \spadfun{PolynomialFactorizationExplicit} category
++ (see \spadfun{factorSquareFreePolynomial}).
randomR: -> R -- has to be global, since has alternative definitions
++ randomR() produces a random element of R
factorSFBRlcUnit: (SupS) -> Factored SupS
++ factorSFBRlcUnit(p) returns the square free factorization of
++ polynomial p
++ (see \spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate})
++ in the case where the leading coefficient of p
++ is a unit.
CODE ==> add
supR: SparseUnivariatePolynomial R
pp: SupS
lpolys,factors: List SupS
r:R
lr:List R
import FactoredFunctionUtilities(SupS)
import FactoredFunctions2(SupR,SupS)
import FactoredFunctions2(S,SupS)
import UnivariatePolynomialCategoryFunctions2(S,SupS,R,SupR)
import UnivariatePolynomialCategoryFunctions2(R,SupR,S,SupS)
-- local function declarations
raise: SupR -> SupS
lower: SupS -> SupR
factorSFBRlcUnitInner: (SupS,R) -> Union(Factored SupS,"failed")
hensel: (SupS,R,List SupS) ->
Union(Record(fctrs:List SupS),"failed")
chooseFSQViableSubstitutions: (SupS) ->
Record(substnsField:R,ppRField:SupR)
--++ chooseFSQViableSubstitutions(p), p is a sup
--++ ("sparse univariate polynomial")
--++ over a sup over R, returns a record
--++ \spad{[substnsField: r, ppRField: q]} where r is a substitution point
--++ q is a sup over R so that the (implicit) variable in q
--++ does not drop in degree and remains square-free.
-- here for the moment, until it compiles
-- N.B., we know that R is NOT a FiniteField, since
-- that is meant to have a special implementation, to break the
-- recursion
solveLinearPolynomialEquationByRecursion(lpolys,pp) ==
lhsdeg:="max"/["max"/[degree v for v in coefficients u] for u in lpolys]
rhsdeg:="max"/[degree v for v in coefficients pp]
lhsdeg = 0 =>
lpolysLower:=[lower u for u in lpolys]
answer:List SupS := [0 for u in lpolys]
for i in 0..rhsdeg repeat
ppx:=map((z1:S):R +-> coefficient(z1,i),pp)
zero? ppx => "next"
recAns:= solveLinearPolynomialEquation(lpolysLower,ppx)
recAns case "failed" => return "failed"
answer:=[monomial(1,i)$S * raise c + d
for c in recAns for d in answer]
answer
solveLinearPolynomialEquationByFractions(lpolys,pp)$LPEBFS
-- local function definitions
hensel(pp,r,factors) ==
-- factors is a relatively prime factorization of pp modulo the ideal
-- (x-r), with suitably imposed leading coefficients.
-- This is lifted, without re-combinations, to a factorization
-- return "failed" if this can't be done
origFactors:=factors
totdegree:Integer:=0
proddegree:Integer:=
"max"/[degree(u) for u in coefficients pp]
n:PI:=1
pn:=prime:=monomial(1,1) - r::S
foundFactors:List SupS:=empty()
while (totdegree <= proddegree) repeat
Ecart:=(pp-*/factors) exquo pn
Ecart case "failed" =>
error "failed lifting in hensel in PFBRU"
zero? Ecart =>
-- then we have all the factors
return [append(foundFactors, factors)]
step:=solveLinearPolynomialEquation(origFactors,
map(z1 +-> elt(z1,r::S),
Ecart))
step case "failed" => return "failed" -- must be a false split
factors:=[a+b*pn for a in factors for b in step]
for a in factors for c in origFactors repeat
pp1:= pp exquo a
pp1 case "failed" => "next"
pp:=pp1
proddegree := proddegree - "max"/[degree(u)
for u in coefficients a]
factors:=remove(a,factors)
origFactors:=remove(c,origFactors)
foundFactors:=[a,:foundFactors]
#factors < 2 =>
return [(empty? factors => foundFactors;
[pp,:foundFactors])]
totdegree:= +/["max"/[degree(u)
for u in coefficients u1]
for u1 in factors]
n:=n+1
pn:=pn*prime
"failed" -- must have been a false split
chooseFSQViableSubstitutions(pp) ==
substns:R
ppR: SupR
while true repeat
substns:= randomR()
zero? elt(leadingCoefficient pp,substns ) => "next"
ppR:=map(z1 +-> elt(z1,substns),pp)
degree gcd(ppR,differentiate ppR)>0 => "next"
leave
[substns,ppR]
raise(supR) == map(z1 +-> z1:R::S,supR)
lower(pp) == map(z1 +-> retract(z1)::R,pp)
factorSFBRlcUnitInner(pp,r) ==
-- pp is square-free as a Sup, but the Up variable occurs.
-- Furthermore, its LC is a unit
-- returns "failed" if the substitution is bad, else a factorization
ppR:=map(z1 +-> elt(z1,r),pp)
degree ppR < degree pp => "failed"
degree gcd(ppR,differentiate ppR) >0 => "failed"
factors:=
fDown:=factorSquareFreePolynomial ppR
[raise (unit fDown * factorList(fDown).first.fctr),
:[raise u.fctr for u in factorList(fDown).rest]]
#factors = 1 => makeFR(1,[["irred",pp,1]])
hen:=hensel(pp,r,factors)
hen case "failed" => "failed"
makeFR(1,[["irred",u,1] for u in hen.fctrs])
-- exported function definitions
if R has StepThrough then
factorSFBRlcUnit(pp) ==
val:R := init()
while true repeat
tempAns:=factorSFBRlcUnitInner(pp,val)
not (tempAns case "failed") => return tempAns
val1:=nextItem val
val1 case "failed" =>
error "at this point, we know we have a finite field"
val:=val1
else
factorSFBRlcUnit(pp) ==
val:R := randomR()
while true repeat
tempAns:=factorSFBRlcUnitInner(pp,val)
not (tempAns case "failed") => return tempAns
val := randomR()
if R has StepThrough then
randomCount:R:= init()
randomR() ==
v:=nextItem(randomCount)
v case "failed" =>
SAY$Lisp "Taking another set of random values"
randomCount:=init()
randomCount
randomCount:=v
randomCount
else if R has random: -> R then
randomR() == random()
else
randomR() == (random()$Integer rem 100)::R
factorByRecursion pp ==
and/[zero? degree u for u in coefficients pp] =>
map(raise,factorPolynomial lower pp)
c:=content pp
unit? c => refine(squareFree pp,factorSquareFreeByRecursion)
pp:=(pp exquo c)::SupS
mergeFactors(refine(squareFree pp,factorSquareFreeByRecursion),
map(z1 +-> z1:S::SupS,factor(c)$S))
factorSquareFreeByRecursion pp ==
and/[zero? degree u for u in coefficients pp] =>
map(raise,factorSquareFreePolynomial lower pp)
unit? (lcpp := leadingCoefficient pp) => factorSFBRlcUnit(pp)
oldnfact:NonNegativeInteger:= 999999
-- I hope we never have to factor a polynomial
-- with more than this number of factors
lcppPow:S
while true repeat -- a loop over possible false splits
cVS:=chooseFSQViableSubstitutions(pp)
newppR:=primitivePart cVS.ppRField
factorsR:=factorSquareFreePolynomial(newppR)
(nfact:=numberOfFactors factorsR) = 1 =>
return makeFR(1,[["irred",pp,1]])
-- OK, force all leading coefficients to be equal to the leading
-- coefficient of the input
nfact > oldnfact => "next" -- can't be a good reduction
oldnfact:=nfact
lcppR:=leadingCoefficient cVS.ppRField
factors:=[raise((lcppR exquo leadingCoefficient u.fctr) ::R * u.fctr)
for u in factorList factorsR]
-- factors now multiplies to give cVS.ppRField * lcppR^(#factors-1)
-- Now change the leading coefficient to be lcpp
factors:=[monomial(lcpp,degree u) + reductum u for u in factors]
ppAdjust:=(lcppPow:=lcpp**#(rest factors)) * pp
OK:=true
hen:=hensel(ppAdjust,cVS.substnsField,factors)
hen case "failed" => "next"
factors:=hen.fctrs
leave
factors:=[ (lc:=content w;
lcppPow:=(lcppPow exquo lc)::S;
(w exquo lc)::SupS)
for w in factors]
not unit? lcppPow =>
error "internal error in factorSquareFreeByRecursion"
makeFR((recip lcppPow)::S::SupS,
[["irred",w,1] for w in factors])
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