/usr/lib/open-axiom/src/algebra/twofact.spad is in open-axiom-source 1.4.1+svn~2626-2ubuntu2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 | --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.
)abbrev package NORMRETR NormRetractPackage
++ Description:
++ This package \undocumented
NormRetractPackage(F, ExtF, SUEx, ExtP, n):C == T where
F : FiniteFieldCategory
ExtF : FiniteAlgebraicExtensionField(F)
SUEx : UnivariatePolynomialCategory ExtF
ExtP : UnivariatePolynomialCategory SUEx
n : PositiveInteger
SUP ==> SparseUnivariatePolynomial
R ==> SUP F
P ==> SUP R
C ==> with
normFactors : ExtP -> List ExtP
++ normFactors(x) \undocumented
retractIfCan : ExtP -> Union(P, "failed")
++ retractIfCan(x) \undocumented
Frobenius : ExtP -> ExtP
++ Frobenius(x) \undocumented
T ==> add
normFactors(p:ExtP):List ExtP ==
facs : List ExtP := [p]
for i in 1..n-1 repeat
member?((p := Frobenius p), facs) => return facs
facs := cons(p, facs)
facs
Frobenius(ff:ExtP):ExtP ==
fft:ExtP:=0
while not zero? ff repeat
fft:=fft + monomial(map(Frobenius, leadingCoefficient ff),
degree ff)
ff:=reductum ff
fft
retractIfCan(ff:ExtP):Union(P, "failed") ==
fft:P:=0
while ff ~= 0 repeat
lc : SUEx := leadingCoefficient ff
plc: SUP F := 0
while lc ~= 0 repeat
lclc:ExtF := leadingCoefficient lc
(retlc := retractIfCan lclc) case "failed" => return "failed"
plc := plc + monomial(retlc::F, degree lc)
lc := reductum lc
fft:=fft+monomial(plc, degree ff)
ff:=reductum ff
fft
)abbrev package TWOFACT TwoFactorize
++ Authors : P.Gianni, J.H.Davenport
++ Date Created : May 1990
++ Date Last Updated: March 1992
++ Description:
++ A basic package for the factorization of bivariate polynomials
++ over a finite field.
++ The functions here represent the base step for the multivariate factorizer.
TwoFactorize(F) : C == T
where
F : FiniteFieldCategory
SUP ==> SparseUnivariatePolynomial
R ==> SUP F
P ==> SUP R
UPCF2 ==> UnivariatePolynomialCategoryFunctions2
C == with
generalTwoFactor : (P) -> Factored P
++ generalTwoFactor(p) returns the factorisation of polynomial p,
++ a sparse univariate polynomial (sup) over a
++ sup over F.
generalSqFr : (P) -> Factored P
++ generalSqFr(p) returns the square-free factorisation of polynomial p,
++ a sparse univariate polynomial (sup) over a
++ sup over F.
twoFactor : (P,Integer) -> Factored P
++ twoFactor(p,n) returns the factorisation of polynomial p,
++ a sparse univariate polynomial (sup) over a
++ sup over F.
++ Also, p is assumed primitive and square-free and n is the
++ degree of the inner variable of p (maximum of the degrees
++ of the coefficients of p).
T == add
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
import CommuteUnivariatePolynomialCategory(F,R,P)
---- Local Functions ----
computeDegree : (P,Integer,Integer) -> PI
exchangeVars : P -> P
exchangeVarTerm: (R, NNI) -> P
pthRoot : (R, NNI, NNI) -> R
-- compute the degree of the extension to reduce the polynomial to a
-- univariate one
computeDegree(m : P,mx:Integer,q:Integer): PI ==
my:=degree m
n1:Integer:=length(10*mx*my)
n2:Integer:=length(q)-1
n:=(n1 quo n2)+1
n::PI
-- n=1 => 1$PositiveInteger
-- (nextPrime(max(n,min(mx,my)))$IntegerPrimesPackage(Integer))::PI
exchangeVars(p : P) : P ==
p = 0 => 0
exchangeVarTerm(leadingCoefficient p, degree p) +
exchangeVars(reductum p)
exchangeVarTerm(c:R, e:NNI) : P ==
c = 0 => 0
monomial(monomial(leadingCoefficient c, e)$R, degree c)$P +
exchangeVarTerm(reductum c, e)
pthRoot(poly:R,p:NonNegativeInteger,PthRootPow:NonNegativeInteger):R ==
tmp:=divideExponents(map((#1::F)**PthRootPow,poly),p)
tmp case "failed" => error "consistency error in TwoFactor"
tmp
fUnion ==> Union("nil", "sqfr", "irred", "prime")
FF ==> Record(flg:fUnion, fctr:P, xpnt:Integer)
generalSqFr(m:P): Factored P ==
m = 0 => 0
degree m = 0 =>
l:=squareFree(leadingCoefficient m)
makeFR(unit(l)::P,[[u.flg,u.fctr::P,u.xpnt] for u in factorList l])
cont := content m
m := (m exquo cont)::P
sqfrm := squareFree m
pfaclist : List FF := empty()
unitPart := unit sqfrm
for u in factorList sqfrm repeat
u.flg = "nil" =>
uexp:NNI:=(u.xpnt):NNI
nfacs:=squareFree(exchangeVars u.fctr)
for v in factorList nfacs repeat
pfaclist:=cons([v.flg, exchangeVars v.fctr, v.xpnt*uexp],
pfaclist)
unitPart := unit(nfacs)**uexp * unitPart
pfaclist := cons(u,pfaclist)
not one? cont =>
sqp := squareFree cont
contlist:=[[w.flg,(w.fctr)::P,w.xpnt] for w in factorList sqp]
pfaclist:= append(contlist, pfaclist)
makeFR(unit(sqp)*unitPart,pfaclist)
makeFR(unitPart,pfaclist)
generalTwoFactor(m:P): Factored P ==
m = 0 => 0
degree m = 0 =>
l:=factor(leadingCoefficient m)$DistinctDegreeFactorize(F,R)
makeFR(unit(l)::P,[[u.flg,u.fctr::P,u.xpnt] for u in factorList l])
ll:List FF
ll:=[]
unitPart:P
cont:=content m
if positive? degree(cont) then
m1:=m exquo cont
m1 case "failed" => error "content doesn't divide"
m:=m1
contfact:=factor(cont)$DistinctDegreeFactorize(F,R)
unitPart:=(unit contfact)::P
ll:=[[w.flg,(w.fctr)::P,w.xpnt] for w in factorList contfact]
else
unitPart:=cont::P
sqfrm:=squareFree m
for u in factors sqfrm repeat
expo:=u.exponent
if negative? expo then error "negative exponent in a factorisation"
expon:NonNegativeInteger:=expo::NonNegativeInteger
fac:=u.factor
degree fac = 1 => ll:=[["irred",fac,expon],:ll]
differentiate fac = 0 =>
-- the polynomial is inseparable w.r.t. its main variable
map(differentiate,fac) = 0 =>
p:=characteristic$F
PthRootPow:=(size()$F exquo p)::NonNegativeInteger
m1:=divideExponents(map(pthRoot(#1,p,PthRootPow),fac),p)
m1 case "failed" => error "consistency error in TwoFactor"
res:=generalTwoFactor m1
unitPart:=unitPart*unit(res)**((p*expon)::NNI)
ll:=[:[[v.flg,v.fctr,expon *p*v.xpnt] for v in factorList res],:ll]
m2:=generalTwoFactor swap fac
unitPart:=unitPart*unit(m2)**(expon::NNI)
ll:=[:[[v.flg,swap v.fctr,expon*v.xpnt] for v in factorList m2],:ll]
ydeg:="max"/[degree w for w in coefficients fac]
twoF:=twoFactor(fac,ydeg)
unitPart:=unitPart*unit(twoF)**expon
ll:=[:[[v.flg,v.fctr,expon*v.xpnt] for v in factorList twoF],
:ll]
makeFR(unitPart,ll)
-- factorization of a primitive square-free bivariate polynomial --
twoFactor(m:P,dx:Integer):Factored P ==
-- choose the degree for the extension
n:PI:=computeDegree(m,dx,size()$F)
-- extend the field
-- find the substitution for x
look:Boolean:=true
dm:=degree m
tryCount:Integer:=min(5,size()$F)
i:Integer:=0
lcm := leadingCoefficient m
umv : R
vval : F
while look and i < tryCount repeat
vval := random()$F
i:=i+1
zero? elt(lcm, vval) => "next value"
umv := map(elt(#1,vval), m)$UPCF2(R, P, F, R)
not zero? degree(gcd(umv,differentiate umv)) => "next val"
n := 1
look := false
extField:=FiniteFieldExtension(F,n)
SUEx:=SUP extField
TP:=SparseUnivariatePolynomial SUEx
mm:TP:=0
m1:=m
while not zero? m1 repeat
mm:=mm+monomial(map(coerce,leadingCoefficient m1)$UPCF2(F,R,
extField,SUEx),degree m1)
m1:=reductum m1
lcmm := leadingCoefficient mm
val : extField
umex : SUEx
if not look then
val := vval :: extField
umex := map(coerce, umv)$UPCF2(F, R, extField, SUEx)
while look repeat
val:=random()$extField
i:=i+1
elt(lcmm,val)=0 => "next value"
umex := map(elt(#1,val), mm)$UPCF2(SUEx, TP, extField, SUEx)
not zero? degree(gcd(umex,differentiate umex)) => "next val"
look:=false
prime:SUEx:=monomial(1,1)-monomial(val,0)
fumex:=factor(umex)$DistinctDegreeFactorize(extField,SUEx)
lfact1:=factors fumex
#lfact1=1 => primeFactor(m,1)
lfact : List TP :=
[map(coerce,lf.factor)$UPCF2(extField,SUEx,SUEx,TP)
for lf in lfact1]
lfact:=cons(map(coerce,unit fumex)$UPCF2(extField,SUEx,SUEx,TP),
lfact)
import GeneralHenselPackage(SUEx,TP)
dx1:PI:=(dx+1)::PI
lfacth:=completeHensel(mm,lfact,prime,dx1)
lfactk: List P :=[]
Normp := NormRetractPackage(F, extField, SUEx, TP, n)
while not empty? lfacth repeat
ff := first lfacth
lfacth := rest lfacth
if not one?(c:=leadingCoefficient leadingCoefficient ff) then
ff:=((inv c)::SUEx)* ff
not ((ffu:= retractIfCan(ff)$Normp) case "failed") =>
lfactk := cons(ffu::P, lfactk)
normfacs := normFactors(ff)$Normp
lfacth := [g for g in lfacth | not member?(g, normfacs)]
ffn := */normfacs
lfactk:=cons(retractIfCan(ffn)$Normp :: P, lfactk)
*/[primeFactor(ff1,1) for ff1 in lfactk]
|