/usr/share/axiom-20170501/src/algebra/TWOFACT.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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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 | )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) : SIG == CODE where
F : FiniteFieldCategory
SUP ==> SparseUnivariatePolynomial
R ==> SUP F
P ==> SUP R
UPCF2 ==> UnivariatePolynomialCategoryFunctions2
SIG ==> 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).
CODE ==> 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
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((x:F):F+->(x::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)
cont ^= 1 =>
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)$DistinctDegreeFactorize(F,R)
makeFR(unit(l)::P,[[u.flg,u.fctr::P,u.xpnt] for u in factorList l])
ll:List FF
ll:=[]
unitPart:P
cont:=content m
if degree(cont)>0 then
m1:=m exquo cont
m1 case "failed" => error "content doesn't divide"
m:=m1
contfact:=factor(cont)$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 expo < 0 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(x+->pthRoot(x,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()$F)
-- extend the field
-- find the substitution for x
look:Boolean:=true
dm:=degree m
try:Integer:=min(5,size()$F)
i:Integer:=0
lcm := leadingCoefficient m
umv : R
while look and i < try repeat
vval := random()$F
i:=i+1
zero? elt(lcm, vval) => "next value"
umv := map(x +-> elt(x,vval), m)$UPCF2(R, P, F, R)
degree(gcd(umv,differentiate umv))^=0 => "next val"
n := 1
look := false
extField:=FiniteFieldExtension(F,n)
SUEx:=SUP extField
TP:=SparseUnivariatePolynomial SUEx
mm:TP:=0
m1:=m
while m1^=0 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()$extField
i:=i+1
elt(lcmm,val)=0 => "next value"
umex := map(x +-> elt(x,val), mm)$UPCF2(SUEx, TP, extField, SUEx)
degree(gcd(umex,differentiate umex))^=0 => "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)$UPCF2(extField,SUEx,SUEx,TP),
lfact)
import GeneralHenselPackage(SUEx,TP)
dx1:PI:=(dx+1)::PI
lfacth:=completeHensel(mm,lfact,prime,dx1)
lfactk: List P :=[]
Normp := NormRetractPackage(F, extField, SUEx, TP, n)
while not empty? lfacth repeat
ff := first lfacth
lfacth := rest lfacth
if (c:=leadingCoefficient leadingCoefficient ff) ^=1 then
ff:=((inv c)::SUEx)* ff
not ((ffu:= retractIfCan(ff)$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]
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