/usr/share/axiom-20170501/src/algebra/PFBR.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 | )abbrev package PFBR PolynomialFactorizationByRecursion
++ Description:
++ PolynomialFactorizationByRecursion(R,E,VarSet,S)
++ is used for factorization of sparse univariate polynomials over
++ a domain S of multivariate polynomials over R.
PolynomialFactorizationByRecursion(R,E, VarSet, S) : SIG == CODE where
R : PolynomialFactorizationExplicit
E : OrderedAbelianMonoidSup
VarSet : OrderedSet
S : PolynomialCategory(R,E,VarSet)
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
bivariateSLPEBR : (List SupS, SupS, VarSet) -> Union(List SupS,"failed")
++ bivariateSLPEBR(lp,p,v) implements
++ the bivariate case of solveLinearPolynomialEquationByRecursion
++ its implementation depends on R
factorSFBRlcUnit : (List VarSet, 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
vv:VarSet
lvpolys,lvpp: List VarSet
r:R
lr:List R
import FactoredFunctionUtilities(SupS)
import FactoredFunctions2(S,SupS)
import FactoredFunctions2(SupR,SupS)
import CommuteUnivariatePolynomialCategory(S,SupS, SupSupS)
import UnivariatePolynomialCategoryFunctions2(S,SupS,SupS,SupSupS)
import UnivariatePolynomialCategoryFunctions2(SupS,SupSupS,S,SupS)
import UnivariatePolynomialCategoryFunctions2(S,SupS,R,SupR)
import UnivariatePolynomialCategoryFunctions2(R,SupR,S,SupS)
import UnivariatePolynomialCategoryFunctions2(S,SupS,SupR,SupSupR)
import UnivariatePolynomialCategoryFunctions2(SupR,SupSupR,S,SupS)
hensel: (SupS,VarSet,R,List SupS) ->
Union(Record(fctrs:List SupS),"failed")
chooseSLPEViableSubstitutions: (List VarSet,List SupS,SupS) ->
Record(substnsField:List R,lpolysRField:List SupR,ppRField:SupR)
--++ chooseSLPEViableSubstitutions(lv,lp,p) chooses substitutions
--++ for the variables in first arg (which are all
--++ the variables that exist) so that the polys in second argument don't
--++ drop in degree and remain square-free, and third arg doesn't drop
--++ drop in degree
chooseFSQViableSubstitutions: (List VarSet,SupS) ->
Record(substnsField:List R,ppRField:SupR)
--++ chooseFSQViableSubstitutions(lv,p) chooses substitutions
--++ for the variables in first arg (which are all
--++ the variables that exist) so that the second argument poly doesn't
--++ drop in degree and remains square-free
raise: SupR -> SupS
lower: SupS -> SupR
SLPEBR: (List SupS, List VarSet, SupS, List VarSet) ->
Union(List SupS,"failed")
factorSFBRlcUnitInner: (List VarSet, SupS,R) ->
Union(Factored SupS,"failed")
hensel(pp,vv,r,factors) ==
origFactors:=factors
totdegree:Integer:=0
proddegree:Integer:=
"max"/[degree(u,vv) for u in coefficients pp]
n:PI:=1
prime:=vv::S - r::S
foundFactors:List SupS:=empty()
while (totdegree <= proddegree) repeat
pn:=prime**n
Ecart:=(pp-*/factors) exquo pn
Ecart case "failed" =>
error "failed lifting in hensel in PFBR"
zero? Ecart =>
-- then we have all the factors
return [append(foundFactors, factors)]
step:=solveLinearPolynomialEquation(origFactors,
map(z1 +-> eval(z1,vv,r),
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,vv)
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,vv)
for u in coefficients u1]
for u1 in factors]
n:=n+1
"failed" -- must have been a false split
factorSFBRlcUnitInner(lvpp,pp,r) ==
-- pp is square-free as a Sup, and its coefficients have precisely
-- the variables of lvpp. Furthermore, its LC is a unit
-- returns "failed" if the substitution is bad, else a factorization
ppR:=map(z1 +-> eval(z1,first lvpp,r),pp)
degree ppR < degree pp => "failed"
degree gcd(ppR,differentiate ppR) >0 => "failed"
factors:=
empty? rest lvpp =>
fDown:=factorSquareFreePolynomial map(z1 +-> retract(z1)::R,ppR)
[raise (unit fDown * factorList(fDown).first.fctr),
:[raise u.fctr for u in factorList(fDown).rest]]
fSame:=factorSFBRlcUnit(rest lvpp,ppR)
[unit fSame * factorList(fSame).first.fctr,
:[uu.fctr for uu in factorList(fSame).rest]]
#factors = 1 => makeFR(1,[["irred",pp,1]])
hen:=hensel(pp,first lvpp,r,factors)
hen case "failed" => "failed"
makeFR(1,[["irred",u,1] for u in hen.fctrs])
if R has StepThrough then
factorSFBRlcUnit(lvpp,pp) ==
val:R := init()
while true repeat
tempAns:=factorSFBRlcUnitInner(lvpp,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(lvpp,pp) ==
val:R := randomR()
while true repeat
tempAns:=factorSFBRlcUnitInner(lvpp,pp,val)
not (tempAns case "failed") => return tempAns
val := randomR()
if R has random: -> R then
randomR() == random()
else
randomR() == (random()$Integer)::R
if R has FiniteFieldCategory then
bivariateSLPEBR(lpolys,pp,v) ==
lpolysR:List SupSupR:=[map(univariate,u) for u in lpolys]
ppR: SupSupR:=map(univariate,pp)
ans:=solveLinearPolynomialEquation(lpolysR,ppR)$SupR
ans case "failed" => "failed"
[map(z1 +-> multivariate(z1,v),w) for w in ans]
else
bivariateSLPEBR(lpolys,pp,v) ==
solveLinearPolynomialEquationByFractions(lpolys,pp)$LPEBFS
chooseFSQViableSubstitutions(lvpp,pp) ==
substns:List R
ppR: SupR
while true repeat
substns:= [randomR() for v in lvpp]
zero? eval(leadingCoefficient pp,lvpp,substns ) => "next"
ppR:=map(z1 +->(retract eval(z1,lvpp,substns))::R,pp)
degree gcd(ppR,differentiate ppR)>0 => "next"
leave
[substns,ppR]
chooseSLPEViableSubstitutions(lvpolys,lpolys,pp) ==
substns:List R
lpolysR:List SupR
ppR: SupR
while true repeat
substns:= [randomR() for v in lvpolys]
zero? eval(leadingCoefficient pp,lvpolys,substns ) => "next"
"or"/[zero? eval(leadingCoefficient u,lvpolys,substns)
for u in lpolys] => "next"
lpolysR:=[map(z1 +-> (retract eval(z1,lvpolys,substns))::R,u)
for u in lpolys]
uu:=lpolysR
while not empty? uu repeat
"or"/[ degree(gcd(uu.first,v))>0 for v in uu.rest] => leave
uu:=rest uu
not empty? uu => "next"
leave
ppR:=map(z1 +-> (retract eval(z1,lvpolys,substns))::R,pp)
[substns,lpolysR,ppR]
raise(supR) == map(z1 +-> z1:R::S,supR)
lower(pp) == map(z1 +-> retract(z1)::R,pp)
SLPEBR(lpolys,lvpolys,pp,lvpp) ==
not empty? (m:=setDifference(lvpp,lvpolys)) =>
v:=first m
lvpp:=remove(v,lvpp)
pp1:SupSupS :=swap map(z1 +-> univariate(z1,v),pp)
-- pp1 is mathematically equal to pp, but is in S[z][v]
-- so we wish to operate on all of its coefficients
ans:List SupSupS:= [0 for u in lpolys]
for m in reverse_! monomials pp1 repeat
ans1:=SLPEBR(lpolys,lvpolys,leadingCoefficient m,lvpp)
ans1 case "failed" => return "failed"
d:=degree m
ans:=[monomial(a1,d)+a for a in ans for a1 in ans1]
[map(z1 +-> multivariate(z1,v),swap pp1) for pp1 in ans]
empty? lvpolys =>
lpolysR:List SupR
ppR:SupR
lpolysR:=[map(retract,u) for u in lpolys]
ppR:=map(retract,pp)
ansR:=solveLinearPolynomialEquation(lpolysR,ppR)
ansR case "failed" => return "failed"
[map(z1 +-> z1::S,uu) for uu in ansR]
cVS:=chooseSLPEViableSubstitutions(lvpolys,lpolys,pp)
ansR:=solveLinearPolynomialEquation(cVS.lpolysRField,cVS.ppRField)
ansR case "failed" => "failed"
#lvpolys = 1 => bivariateSLPEBR(lpolys,pp, first lvpolys)
solveLinearPolynomialEquationByFractions(lpolys,pp)$LPEBFS
solveLinearPolynomialEquationByRecursion(lpolys,pp) ==
lvpolys := removeDuplicates_!
concat [ concat [variables z for z in coefficients u]
for u in lpolys]
lvpp := removeDuplicates_!
concat [variables z for z in coefficients pp]
SLPEBR(lpolys,lvpolys,pp,lvpp)
factorByRecursion pp ==
lv:List(VarSet) := removeDuplicates_!
concat [variables z for z in coefficients pp]
empty? lv =>
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 ==
lv:List(VarSet) := removeDuplicates_!
concat [variables z for z in coefficients pp]
empty? lv =>
map(raise,factorPolynomial lower pp)
unit? (lcpp := leadingCoefficient pp) => factorSFBRlcUnit(lv,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
cVS:=chooseFSQViableSubstitutions(lv,pp)
factorsR:=factorSquareFreePolynomial(cVS.ppRField)
(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
factors:=[(lcpp exquo leadingCoefficient u.fctr)::S * raise u.fctr
for u in factorList factorsR]
ppAdjust:=(lcppPow:=lcpp**#(rest factors)) * pp
lvppList:=lv
OK:=true
for u in lvppList for v in cVS.substnsField repeat
hen:=hensel(ppAdjust,u,v,factors)
hen case "failed" =>
OK:=false
"leave"
factors:=hen.fctrs
OK => 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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