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--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
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--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 IBPTOOLS IntegralBasisPolynomialTools
++ Author: Clifton Williamson
++ Date Created: 13 August 1993
++ Date Last Updated: 17 August 1993
++ Basic Operations: mapUnivariate, mapBivariate
++ Related Domains: PAdicWildFunctionFieldIntegralBasis(K,R,UP,F)
++ Also See: WildFunctionFieldIntegralBasis, FunctionFieldIntegralBasis
++ AMS Classifications:
++ Keywords: function field, finite field, integral basis
++ Examples:
++ References:
++ Description: IntegralBasisPolynomialTools provides functions for
++ mapping functions on the coefficients of univariate and bivariate
++ polynomials.
IntegralBasisPolynomialTools(K,R,UP,L): Exports == Implementation where
K : Ring
R : UnivariatePolynomialCategory K
UP : UnivariatePolynomialCategory R
L : Ring
MAT ==> Matrix
SUP ==> SparseUnivariatePolynomial
Exports ==> with
mapUnivariate: (L -> K,SUP L) -> R
++ mapUnivariate(f,p(x)) applies the function \spad{f} to the
++ coefficients of \spad{p(x)}.
mapUnivariate: (K -> L,R) -> SUP L
++ mapUnivariate(f,p(x)) applies the function \spad{f} to the
++ coefficients of \spad{p(x)}.
mapUnivariateIfCan: (L -> Union(K,"failed"),SUP L) -> Union(R,"failed")
++ mapUnivariateIfCan(f,p(x)) applies the function \spad{f} to the
++ coefficients of \spad{p(x)}, if possible, and returns
++ \spad{"failed"} otherwise.
mapMatrixIfCan: (L -> Union(K,"failed"),MAT SUP L) -> Union(MAT R,"failed")
++ mapMatrixIfCan(f,mat) applies the function \spad{f} to the
++ coefficients of the entries of \spad{mat} if possible, and returns
++ \spad{"failed"} otherwise.
mapBivariate: (K -> L,UP) -> SUP SUP L
++ mapBivariate(f,p(x,y)) applies the function \spad{f} to the
++ coefficients of \spad{p(x,y)}.
Implementation ==> add
mapUnivariate(f:L -> K,poly:SUP L) ==
ans : R := 0
while not zero? poly repeat
ans := ans + monomial(f leadingCoefficient poly,degree poly)
poly := reductum poly
ans
mapUnivariate(f:K -> L,poly:R) ==
ans : SUP L := 0
while not zero? poly repeat
ans := ans + monomial(f leadingCoefficient poly,degree poly)
poly := reductum poly
ans
mapUnivariateIfCan(f,poly) ==
ans : R := 0
while not zero? poly repeat
(lc := f leadingCoefficient poly) case "failed" => return "failed"
ans := ans + monomial(lc :: K,degree poly)
poly := reductum poly
ans
mapMatrixIfCan(f,mat) ==
m := nrows mat; n := ncols mat
matOut : MAT R := new(m,n,0)
for i in 1..m repeat for j in 1..n repeat
(poly := mapUnivariateIfCan(f,qelt(mat,i,j))) case "failed" =>
return "failed"
qsetelt!(matOut,i,j,poly :: R)
matOut
mapBivariate(f,poly) ==
ans : SUP SUP L := 0
while not zero? poly repeat
ans :=
ans + monomial(mapUnivariate(f,leadingCoefficient poly),degree poly)
poly := reductum poly
ans
)abbrev package IBACHIN ChineseRemainderToolsForIntegralBases
++ Author: Clifton Williamson
++ Date Created: 9 August 1993
++ Date Last Updated: 3 December 1993
++ Basic Operations: chineseRemainder, factorList
++ Related Domains: PAdicWildFunctionFieldIntegralBasis(K,R,UP,F)
++ Also See: WildFunctionFieldIntegralBasis, FunctionFieldIntegralBasis
++ AMS Classifications:
++ Keywords: function field, finite field, integral basis
++ Examples:
++ References:
++ Description:
ChineseRemainderToolsForIntegralBases(K,R,UP): Exports == Implementation where
K : FiniteFieldCategory
R : UnivariatePolynomialCategory K
UP : UnivariatePolynomialCategory R
DDFACT ==> DistinctDegreeFactorize
I ==> Integer
L ==> List
L2 ==> ListFunctions2
Mat ==> Matrix R
NNI ==> NonNegativeInteger
PI ==> PositiveInteger
Q ==> Fraction R
SAE ==> SimpleAlgebraicExtension
SUP ==> SparseUnivariatePolynomial
SUP2 ==> SparseUnivariatePolynomialFunctions2
Result ==> Record(basis: Mat, basisDen: R, basisInv: Mat)
Exports ==> with
factorList: (K,NNI,NNI,NNI) -> L SUP K
++ factorList(k,n,m,j) \undocumented
listConjugateBases: (Result,NNI,NNI) -> List Result
++ listConjugateBases(bas,q,n) returns the list
++ \spad{[bas,bas^Frob,bas^(Frob^2),...bas^(Frob^(n-1))]}, where
++ \spad{Frob} raises the coefficients of all polynomials
++ appearing in the basis \spad{bas} to the \spad{q}th power.
chineseRemainder: (List UP, List Result, NNI) -> Result
++ chineseRemainder(lu,lr,n) \undocumented
Implementation ==> add
import ModularHermitianRowReduction(R)
import TriangularMatrixOperations(R, Vector R, Vector R, Matrix R)
applyFrobToMatrix: (Matrix R,NNI) -> Matrix R
applyFrobToMatrix(mat,q) ==
-- raises the coefficients of the polynomial entries of 'mat'
-- to the qth power
m := nrows mat; n := ncols mat
ans : Matrix R := new(m,n,0)
for i in 1..m repeat for j in 1..n repeat
qsetelt!(ans,i,j,map(#1 ** q,qelt(mat,i,j)))
ans
listConjugateBases(bas,q,n) ==
outList : List Result := list bas
b := bas.basis; bInv := bas.basisInv; bDen := bas.basisDen
for i in 1..(n-1) repeat
b := applyFrobToMatrix(b,q)
bInv := applyFrobToMatrix(bInv,q)
bDen := map(#1 ** q,bDen)
newBasis : Result := [b,bDen,bInv]
outList := concat(newBasis,outList)
reverse! outList
factorList(a,q,n,k) ==
coef : SUP K := monomial(a,0); xx : SUP K := monomial(1,1)
outList : L SUP K := list((xx - coef)**k)
for i in 1..(n-1) repeat
coef := coef ** q
outList := concat((xx - coef)**k,outList)
reverse! outList
basisInfoToPolys: (Mat,R,R) -> L UP
basisInfoToPolys(mat,lcm,den) ==
n := nrows(mat) :: I; n1 := n - 1
outList : L UP := empty()
for i in 1..n repeat
pp : UP := 0
for j in 0..n1 repeat
pp := pp + monomial((lcm quo den) * qelt(mat,i,j+1),j)
outList := concat(pp,outList)
reverse! outList
basesToPolyLists: (L Result,R) -> L L UP
basesToPolyLists(basisList,lcm) ==
[basisInfoToPolys(b.basis,lcm,b.basisDen) for b in basisList]
OUT ==> OutputForm
approximateExtendedEuclidean: (UP,UP,R,NNI) -> Record(coef1:UP,coef2:UP)
approximateExtendedEuclidean(f,g,p,n) ==
-- f and g are monic and relatively prime (mod p)
-- function returns [coef1,coef2] such that
-- coef1 * f + coef2 * g = 1 (mod p^n)
sae := SAE(K,R,p)
fSUP : SUP R := makeSUP f; gSUP : SUP R := makeSUP g
fBar : SUP sae := map(convert(#1)@sae,fSUP)$SUP2(R,sae)
gBar : SUP sae := map(convert(#1)@sae,gSUP)$SUP2(R,sae)
ee := extendedEuclidean(fBar,gBar)
not one?(ee.generator) =>
error "polynomials aren't relatively prime"
ss1 := ee.coef1; tt1 := ee.coef2
s1 : SUP R := map(convert(#1)@R,ss1)$SUP2(sae,R); s := s1
t1 : SUP R := map(convert(#1)@R,tt1)$SUP2(sae,R); t := t1
pPower := p
for i in 2..n repeat
num := 1 - s * fSUP - t * gSUP
rhs := (num exquo pPower) :: SUP R
sigma := map(#1 rem p,s1 * rhs); tau := map(#1 rem p,t1 * rhs)
s := s + pPower * sigma; t := t + pPower * tau
quorem := monicDivide(s,gSUP)
pPower := pPower * p
s := map(#1 rem pPower,quorem.remainder)
t := map(#1 rem pPower,t + fSUP * (quorem.quotient))
[unmakeSUP s,unmakeSUP t]
--mapChineseToList: (L SUP Q,L SUP Q,I) -> L SUP Q
--mapChineseToList(list,polyList,i) ==
mapChineseToList: (L UP,L UP,I,R) -> L UP
mapChineseToList(list,polyList,i,den) ==
-- 'polyList' consists of MONIC polynomials
-- computes a polynomial p such that p = pp (modulo polyList[i])
-- and p = 0 (modulo polyList[j]) for j ~= i for each 'pp' in 'list'
-- create polynomials
q : UP := 1
for j in 1..(i-1) repeat
q := q * first polyList
polyList := rest polyList
p := first polyList
polyList := rest polyList
for j in (i+1).. while not empty? polyList repeat
q := q * first polyList
polyList := rest polyList
--p := map((numer(#1) rem den)/1, p)
--q := map((numer(#1) rem den)/1, q)
-- 'den' is a power of an irreducible polynomial
--!! make this computation more efficient!!
factoredDen := factor(den)$DDFACT(K,R)
prime := nthFactor(factoredDen,1)
n := nthExponent(factoredDen,1) :: NNI
invPoly := approximateExtendedEuclidean(q,p,prime,n).coef1
-- monicDivide may be inefficient?
[monicDivide(pp * invPoly * q,p * q).remainder for pp in list]
polyListToMatrix: (L UP,NNI) -> Mat
polyListToMatrix(polyList,n) ==
mat : Mat := new(n,n,0)
for i in 1..n for poly in polyList repeat
while not zero? poly repeat
mat(i,degree(poly) + 1) := leadingCoefficient poly
poly := reductum poly
mat
chineseRemainder(factors,factorBases,n) ==
denLCM : R := reduce("lcm",[base.basisDen for base in factorBases])
denLCM = 1 => [scalarMatrix(n,1),1,scalarMatrix(n,1)]
-- compute local basis polynomials with denominators cleared
factorBasisPolyLists := basesToPolyLists(factorBases,denLCM)
-- use Chinese remainder to compute basis polynomials w/o denominators
basisPolyLists : L L UP := empty()
for i in 1.. for pList in factorBasisPolyLists repeat
polyList := mapChineseToList(pList,factors,i,denLCM)
basisPolyLists := concat(polyList,basisPolyLists)
basisPolys := concat reverse! basisPolyLists
mat := squareTop rowEchelon(polyListToMatrix(basisPolys,n),denLCM)
matInv := UpTriBddDenomInv(mat,denLCM)
[mat,denLCM,matInv]
)abbrev package PWFFINTB PAdicWildFunctionFieldIntegralBasis
++ Author: Clifton Williamson
++ Date Created: 5 July 1993
++ Date Last Updated: 17 August 1993
++ Basic Operations: integralBasis, localIntegralBasis
++ Related Domains: WildFunctionFieldIntegralBasis(K,R,UP,F)
++ Also See: FunctionFieldIntegralBasis
++ AMS Classifications:
++ Keywords: function field, finite field, integral basis
++ Examples:
++ References:
++ Description:
++ In this package K is a finite field, R is a ring of univariate
++ polynomials over K, and F is a monogenic algebra over R.
++ We require that F is monogenic, i.e. that \spad{F = K[x,y]/(f(x,y))},
++ because the integral basis algorithm used will factor the polynomial
++ \spad{f(x,y)}. The package provides a function to compute the integral
++ closure of R in the quotient field of F as well as a function to compute
++ a "local integral basis" at a specific prime.
PAdicWildFunctionFieldIntegralBasis(K,R,UP,F): Exports == Implementation where
K : FiniteFieldCategory
R : UnivariatePolynomialCategory K
UP : UnivariatePolynomialCategory R
F : MonogenicAlgebra(R,UP)
I ==> Integer
L ==> List
L2 ==> ListFunctions2
Mat ==> Matrix R
NNI ==> NonNegativeInteger
PI ==> PositiveInteger
Q ==> Fraction R
SAE ==> SimpleAlgebraicExtension
SUP ==> SparseUnivariatePolynomial
CDEN ==> CommonDenominator
DDFACT ==> DistinctDegreeFactorize
WFFINTBS ==> WildFunctionFieldIntegralBasis
Result ==> Record(basis: Mat, basisDen: R, basisInv:Mat)
IResult ==> Record(basis: Mat, basisDen: R, basisInv:Mat,discr: R)
IBPTOOLS ==> IntegralBasisPolynomialTools
IBACHIN ==> ChineseRemainderToolsForIntegralBases
IRREDFFX ==> IrredPolyOverFiniteField
GHEN ==> GeneralHenselPackage
Exports ==> with
integralBasis : () -> Result
++ \spad{integralBasis()} returns a record
++ \spad{[basis,basisDen,basisInv] } containing information regarding
++ the integral closure of R in the quotient field of the framed
++ algebra F. F is a framed algebra with R-module basis
++ \spad{w1,w2,...,wn}.
++ If 'basis' is the matrix \spad{(aij, i = 1..n, j = 1..n)}, then
++ the \spad{i}th element of the integral basis is
++ \spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)}, i.e. the
++ \spad{i}th row of 'basis' contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix 'basisInv' contains the coordinates of \spad{wi} with respect
++ to the basis \spad{v1,...,vn}: if 'basisInv' is the matrix
++ \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
localIntegralBasis : R -> Result
++ \spad{integralBasis(p)} returns a record
++ \spad{[basis,basisDen,basisInv] } containing information regarding
++ the local integral closure of R at the prime \spad{p} in the quotient
++ field of the framed algebra F. F is a framed algebra with R-module
++ basis \spad{w1,w2,...,wn}.
++ If 'basis' is the matrix \spad{(aij, i = 1..n, j = 1..n)}, then
++ the \spad{i}th element of the local integral basis is
++ \spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)}, i.e. the
++ \spad{i}th row of 'basis' contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix 'basisInv' contains the coordinates of \spad{wi} with respect
++ to the basis \spad{v1,...,vn}: if 'basisInv' is the matrix
++ \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
reducedDiscriminant: UP -> R
++ reducedDiscriminant(up) \undocumented
Implementation ==> add
import IntegralBasisTools(R, UP, F)
import GeneralHenselPackage(R,UP)
import ModularHermitianRowReduction(R)
import TriangularMatrixOperations(R, Vector R, Vector R, Matrix R)
reducedDiscriminant f ==
ff : SUP Q := mapUnivariate(#1 :: Q,f)$IBPTOOLS(R,UP,SUP UP,Q)
ee := extendedEuclidean(ff,differentiate ff)
cc := concat(coefficients(ee.coef1),coefficients(ee.coef2))
cden := splitDenominator(cc)$CDEN(R,Q,L Q)
denom := cden.den
gg := gcd map(numer,cden.num)$L2(Q,R)
(ans := denom exquo gg) case "failed" =>
error "PWFFINTB: error in reduced discriminant computation"
ans :: R
compLocalBasis: (UP,R) -> Result
compLocalBasis(poly,prime) ==
-- compute a local integral basis at 'prime' for k[x,y]/(poly(x,y)).
sae := SAE(R,UP,poly)
localIntegralBasis(prime)$WFFINTBS(K,R,UP,sae)
compLocalBasisOverExt: (UP,R,UP,NNI) -> Result
compLocalBasisOverExt(poly0,prime0,irrPoly0,k) ==
-- poly0 = irrPoly0**k (mod prime0)
n := degree poly0; disc0 := discriminant poly0
(disc0 exquo prime0) case "failed" =>
[scalarMatrix(n,1), 1, scalarMatrix(n,1)]
r := degree irrPoly0
-- extend scalars:
-- construct irreducible polynomial of degree r over K
irrPoly := generateIrredPoly(r :: PI)$IRREDFFX(K)
-- construct extension of degree r over K
E := SAE(K,SUP K,irrPoly)
-- lift coefficients to elements of E
poly := mapBivariate(#1 :: E,poly0)$IBPTOOLS(K,R,UP,E)
redDisc0 := reducedDiscriminant poly0
redDisc := mapUnivariate(#1 :: E,redDisc0)$IBPTOOLS(K,R,UP,E)
prime := mapUnivariate(#1 :: E,prime0)$IBPTOOLS(K,R,UP,E)
sae := SAE(E,SUP E,prime)
-- reduction (mod prime) of polynomial of which poly is the kth power
redIrrPoly :=
pp := mapBivariate(#1 :: E,irrPoly0)$IBPTOOLS(K,R,UP,E)
mapUnivariate(reduce,pp)$IBPTOOLS(SUP E,SUP SUP E,SUP SUP SUP E,sae)
-- factor the reduction
factorListSAE := factors factor(redIrrPoly)$DDFACT(sae,SUP sae)
-- list the 'primary factors' of the reduction of poly
redFactors : List SUP sae := [(f.factor)**k for f in factorListSAE]
-- lift these factors to elements of SUP SUP E
primaries : List SUP SUP E :=
[mapUnivariate(lift,ff)$IBPTOOLS(SUP E,SUP SUP E,SUP SUP SUP E,sae) _
for ff in redFactors]
-- lift the factors to factors modulo a suitable power of 'prime'
deg := (1 + order(redDisc,prime) * degree(prime)) :: PI
henselInfo := HenselLift(poly,primaries,prime,deg)$GHEN(SUP E,SUP SUP E)
henselFactors := henselInfo.plist
psi1 := first henselFactors
FF := SAE(SUP E,SUP SUP E,psi1)
factorIb := localIntegralBasis(prime)$WFFINTBS(E,SUP E,SUP SUP E,FF)
bs := listConjugateBases(factorIb,size()$K,r)$IBACHIN(E,SUP E,SUP SUP E)
ib := chineseRemainder(henselFactors,bs,n)$IBACHIN(E,SUP E,SUP SUP E)
b : Matrix R :=
bas := mapMatrixIfCan(retractIfCan,ib.basis)$IBPTOOLS(K,R,UP,E)
bas case "failed" => error "retraction of basis failed"
bas :: Matrix R
bInv : Matrix R :=
--bas := mapMatrixIfCan(ric,ib.basisInv)$IBPTOOLS(K,R,UP,E)
bas := mapMatrixIfCan(retractIfCan,ib.basisInv)$IBPTOOLS(K,R,UP,E)
bas case "failed" => error "retraction of basis inverse failed"
bas :: Matrix R
bDen : R :=
p := mapUnivariateIfCan(retractIfCan,ib.basisDen)$IBPTOOLS(K,R,UP,E)
p case "failed" => error "retraction of basis denominator failed"
p :: R
[b,bDen,bInv]
padicLocalIntegralBasis: (UP,R,R,R) -> IResult
padicLocalIntegralBasis(p,disc,redDisc,prime) ==
-- polynomials in x modulo 'prime'
sae := SAE(K,R,prime)
-- find the factorization of 'p' modulo 'prime' and lift the
-- prime powers to elements of UP:
-- reduce 'p' modulo 'prime'
reducedP := mapUnivariate(reduce,p)$IBPTOOLS(R,UP,SUP UP,sae)
-- factor the reduced polynomial
factorListSAE := factors factor(reducedP)$DDFACT(sae,SUP sae)
-- if only one prime factor, perform usual integral basis computation
(# factorListSAE) = 1 =>
ib := localIntegralBasis(prime)$WFFINTBS(K,R,UP,F)
index := diagonalProduct(ib.basisInv)
[ib.basis,ib.basisDen,ib.basisInv,disc quo (index * index)]
-- list the 'prime factors' of the reduced polynomial
redPrimes : List SUP sae :=
[f.factor for f in factorListSAE]
-- lift these factors to elements of UP
primes : List UP :=
[mapUnivariate(lift,ff)$IBPTOOLS(R,UP,SUP UP,sae) for ff in redPrimes]
-- list the exponents
expons : List NNI := [((f.exponent) :: NNI) for f in factorListSAE]
-- list the 'primary factors' of the reduced polynomial
redPrimaries : List SUP sae :=
[(f.factor) **((f.exponent) :: NNI) for f in factorListSAE]
-- lift these factors to elements of UP
primaries : List UP :=
[mapUnivariate(lift,ff)$IBPTOOLS(R,UP,SUP UP,sae) for ff in redPrimaries]
-- lift the factors to factors modulo a suitable power of 'prime'
deg := (1 + order(redDisc,prime) * degree(prime)) :: PI
henselInfo := HenselLift(p,primaries,prime,deg)
henselFactors := henselInfo.plist
-- compute integral bases for the factors
factorBases : List Result := empty(); degPrime := degree prime
for pp in primes for k in expons for qq in henselFactors repeat
base :=
degPp := degree pp
degPp > 1 and gcd(degPp,degPrime) = 1 =>
compLocalBasisOverExt(qq,prime,pp,k)
compLocalBasis(qq,prime)
factorBases := concat(base,factorBases)
factorBases := reverse! factorBases
ib := chineseRemainder(henselFactors,factorBases,rank()$F)$IBACHIN(K,R,UP)
index := diagonalProduct(ib.basisInv)
[ib.basis,ib.basisDen,ib.basisInv,disc quo (index * index)]
localIntegralBasis prime ==
p := definingPolynomial()$F; disc := discriminant p
--disc := determinant traceMatrix()$F
redDisc := reducedDiscriminant p
ib := padicLocalIntegralBasis(p,disc,redDisc,prime)
[ib.basis,ib.basisDen,ib.basisInv]
listSquaredFactors: R -> List R
listSquaredFactors px ==
-- returns a list of the factors of px which occur with
-- exponent > 1
ans : List R := empty()
factored := factor(px)$DistinctDegreeFactorize(K,R)
for f in factors(factored) repeat
if f.exponent > 1 then ans := concat(f.factor,ans)
ans
integralBasis() ==
p := definingPolynomial()$F; disc := discriminant p; n := rank()$F
--traceMat := traceMatrix()$F; n := rank()$F
--disc := determinant traceMat -- discriminant of current order
singList := listSquaredFactors disc -- singularities of relative Spec
redDisc := reducedDiscriminant p
runningRb := runningRbinv := scalarMatrix(n,1)$Mat
-- runningRb = basis matrix of current order
-- runningRbinv = inverse basis matrix of current order
-- these are wrt the original basis for F
runningRbden : R := 1
-- runningRbden = denominator for current basis matrix
empty? singList => [runningRb, runningRbden, runningRbinv]
for prime in singList repeat
lb := padicLocalIntegralBasis(p,disc,redDisc,prime)
rb := lb.basis; rbinv := lb.basisInv; rbden := lb.basisDen
disc := lb.discr
mat := vertConcat(rbden * runningRb,runningRbden * rb)
runningRbden := runningRbden * rbden
runningRb := squareTop rowEchelon(mat,runningRbden)
--runningRb := squareTop rowEch mat
runningRbinv := UpTriBddDenomInv(runningRb,runningRbden)
[runningRb, runningRbden, runningRbinv]
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