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
|
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra geneez.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GENEEZ GenExEuclid}
<<package GENEEZ GenExEuclid>>=
)abbrev package GENEEZ GenExEuclid
++ Author : P.Gianni.
++ January 1990
++ The equation \spad{Af+Bg=h} and its generalization to n polynomials
++ is solved for solutions over the R, euclidean domain.
++ A table containing the solutions of \spad{Af+Bg=x**k} is used.
++ The operations are performed modulus a prime which are in principle big enough,
++ but the solutions are tested and, in case of failure, a hensel
++ lifting process is used to get to the right solutions.
++ It will be used in the factorization of multivariate polynomials
++ over finite field, with \spad{R=F[x]}.
GenExEuclid(R,BP) : C == T
where
R : EuclideanDomain
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
BP : UnivariatePolynomialCategory R
L ==> List
C == with
reduction: (BP,R) -> BP
++ reduction(p,prime) reduces the polynomial p modulo prime of R.
++ Note: this function is exported only because it's conditional.
compBound: (BP,L BP) -> NNI
++ compBound(p,lp)
++ computes a bound for the coefficients of the solution
++ polynomials.
++ Given a polynomial right hand side p, and a list lp of left hand side polynomials.
++ Exported because it depends on the valuation.
tablePow : (NNI,R,L BP) -> Union(Vector(L BP),"failed")
++ tablePow(maxdeg,prime,lpol) constructs the table with the
++ coefficients of the Extended Euclidean Algorithm for lpol.
++ Here the right side is \spad{x**k}, for k less or equal to maxdeg.
++ The operation returns "failed" when the elements are not coprime modulo prime.
solveid : (BP,R,Vector L BP) -> Union(L BP,"failed")
++ solveid(h,table) computes the coefficients of the
++ extended euclidean algorithm for a list of polynomials
++ whose tablePow is table and with right side h.
testModulus : (R, L BP) -> Boolean
++ testModulus(p,lp) returns true if the the prime p
++ is valid for the list of polynomials lp, i.e. preserves
++ the degree and they remain relatively prime.
T == add
if R has multiplicativeValuation then
compBound(m:BP,listpolys:L BP) : NNI ==
ldeg:=[degree f for f in listpolys]
n:NNI:= (+/[df for df in ldeg])
normlist:=[ +/[euclideanSize(u)**2 for u in coefficients f]
for f in listpolys]
nm:= +/[euclideanSize(u)**2 for u in coefficients m]
normprod := */[g**((n-df)::NNI) for g in normlist for df in ldeg]
2*(approxSqrt(normprod * nm)$IntegerRoots(Integer))::NNI
else if R has additiveValuation then
-- a fairly crude Hadamard-style bound for the solution
-- based on regarding the problem as a system of linear equations.
compBound(m:BP,listpolys:L BP) : NNI ==
"max"/[euclideanSize u for u in coefficients m] +
+/["max"/[euclideanSize u for u in coefficients p]
for p in listpolys]
else
compBound(m:BP,listpolys:L BP) : NNI ==
error "attempt to use compBound without a well-understood valuation"
if R has IntegerNumberSystem then
reduction(u:BP,p:R):BP ==
p = 0 => u
map(symmetricRemainder(#1,p),u)
else reduction(u:BP,p:R):BP ==
p = 0 => u
map(#1 rem p,u)
merge(p:R,q:R):Union(R,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
modInverse(c:R,p:R):R ==
(extendedEuclidean(c,p,1)::Record(coef1:R,coef2:R)).coef1
exactquo(u:BP,v:BP,p:R):Union(BP,"failed") ==
invlcv:=modInverse(leadingCoefficient v,p)
r:=monicDivide(u,reduction(invlcv*v,p))
reduction(r.remainder,p) ~=0 => "failed"
reduction(invlcv*r.quotient,p)
FP:=EuclideanModularRing(R,BP,R,reduction,merge,exactquo)
--make table global variable!
table:Vector L BP
import GeneralHenselPackage(R,BP)
--local functions
makeProducts : L BP -> L BP
liftSol: (L BP,BP,R,R,Vector L BP,BP,NNI) -> Union(L BP,"failed")
reduceList(lp:L BP,lmod:R): L FP ==[reduce(ff,lmod) for ff in lp]
coerceLFP(lf:L FP):L BP == [fm::BP for fm in lf]
liftSol(oldsol:L BP,err:BP,lmod:R,lmodk:R,
table:Vector L BP,m:BP,bound:NNI):Union(L BP,"failed") ==
euclideanSize(lmodk) > bound => "failed"
d:=degree err
ftab:Vector L FP :=
map(reduceList(#1,lmod),table)$VectorFunctions2(List BP,List FP)
sln:L FP:=[0$FP for xx in ftab.1 ]
for i in 0 .. d |(cc:=coefficient(err,i)) ~=0 repeat
sln:=[slp+reduce(cc::BP,lmod)*pp
for pp in ftab.(i+1) for slp in sln]
nsol:=[f-lmodk*reduction(g::BP,lmod) for f in oldsol for g in sln]
lmodk1:=lmod*lmodk
nsol:=[reduction(slp,lmodk1) for slp in nsol]
lpolys:L BP:=table.(#table)
(fs:=+/[f*g for f in lpolys for g in nsol]) = m => nsol
a:BP:=((fs-m) exquo lmodk1)::BP
liftSol(nsol,a,lmod,lmodk1,table,m,bound)
makeProducts(listPol:L BP):L BP ==
#listPol < 2 => listPol
#listPol = 2 => reverse listPol
f:= first listPol
ll := rest listPol
[*/ll,:[f*g for g in makeProducts ll]]
testModulus(pmod, listPol) ==
redListPol := reduceList(listPol, pmod)
for pol in listPol for rpol in redListPol repeat
degree(pol) ~= degree(rpol::BP) => return false
while not empty? redListPol repeat
rpol := first redListPol
redListPol := rest redListPol
for rpol2 in redListPol repeat
gcd(rpol, rpol2) ~= 1 => return false
true
if R has Field then
tablePow(mdeg:NNI,pmod:R,listPol:L BP) ==
multiE:=multiEuclidean(listPol,1$BP)
multiE case "failed" => "failed"
ptable:Vector L BP :=new(mdeg+1,[])
ptable.1:=multiE
x:BP:=monomial(1,1)
for i in 2..mdeg repeat ptable.i:=
[tpol*x rem fpol for tpol in ptable.(i-1) for fpol in listPol]
ptable.(mdeg+1):=makeProducts listPol
ptable
solveid(m:BP,pmod:R,table:Vector L BP) : Union(L BP,"failed") ==
-- Actually, there's no possibility of failure
d:=degree m
sln:L BP:=[0$BP for xx in table.1]
for i in 0 .. d | coefficient(m,i)~=0 repeat
sln:=[slp+coefficient(m,i)*pp
for pp in table.(i+1) for slp in sln]
sln
else
tablePow(mdeg:NNI,pmod:R,listPol:L BP) ==
listP:L FP:= [reduce(pol,pmod) for pol in listPol]
multiE:=multiEuclidean(listP,1$FP)
multiE case "failed" => "failed"
ftable:Vector L FP :=new(mdeg+1,[])
fl:L FP:= [ff::FP for ff in multiE]
ftable.1:=[fpol for fpol in fl]
x:FP:=reduce(monomial(1,1),pmod)
for i in 2..mdeg repeat ftable.i:=
[tpol*x rem fpol for tpol in ftable.(i-1) for fpol in listP]
ptable:= map(coerceLFP,ftable)$VectorFunctions2(List FP,List BP)
ptable.(mdeg+1):=makeProducts listPol
ptable
solveid(m:BP,pmod:R,table:Vector L BP) : Union(L BP,"failed") ==
d:=degree m
ftab:Vector L FP:=
map(reduceList(#1,pmod),table)$VectorFunctions2(List BP,List FP)
lpolys:L BP:=table.(#table)
sln:L FP:=[0$FP for xx in ftab.1]
for i in 0 .. d | coefficient(m,i)~=0 repeat
sln:=[slp+reduce(coefficient(m,i)::BP,pmod)*pp
for pp in ftab.(i+1) for slp in sln]
soln:=[slp::BP for slp in sln]
(fs:=+/[f*g for f in lpolys for g in soln]) = m=> soln
-- Compute bound
bound:=compBound(m,lpolys)
a:BP:=((fs-m) exquo pmod)::BP
liftSol(soln,a,pmod,pmod,table,m,bound)
@
\section{License}
<<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.
@
<<*>>=
<<license>>
<<package GENEEZ GenExEuclid>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
