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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
--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 TRIMAT TriangularMatrixOperations
++ Fraction free inverses of triangular matrices
++ Author: Victor Miller
++ Date Created:
++ Date Last Updated: 24 Jul 1990
++ Keywords:
++ Examples:
++ References:
++ Description:
++ This package provides functions that compute "fraction-free"
++ inverses of upper and lower triangular matrices over a integral
++ domain. By "fraction-free inverses" we mean the following:
++ given a matrix B with entries in R and an element d of R such that
++ d * inv(B) also has entries in R, we return d * inv(B). Thus,
++ it is not necessary to pass to the quotient field in any of our
++ computations.
TriangularMatrixOperations(R,Row,Col,M): Exports == Implementation where
R : IntegralDomain
Row : FiniteLinearAggregate R
Col : FiniteLinearAggregate R
M : MatrixCategory(R,Row,Col)
Exports ==> with
UpTriBddDenomInv: (M,R) -> M
++ UpTriBddDenomInv(B,d) returns M, where
++ B is a non-singular upper triangular matrix and d is an
++ element of R such that \spad{M = d * inv(B)} has entries in R.
LowTriBddDenomInv:(M,R) -> M
++ LowTriBddDenomInv(B,d) returns M, where
++ B is a non-singular lower triangular matrix and d is an
++ element of R such that \spad{M = d * inv(B)} has entries in R.
Implementation ==> add
UpTriBddDenomInv(A,denom) ==
AI := zero(nrows A, nrows A)$M
offset := minColIndex AI - minRowIndex AI
for i in minRowIndex AI .. maxRowIndex AI
for j in minColIndex AI .. maxColIndex AI repeat
qsetelt!(AI,i,j,(denom exquo qelt(A,i,j))::R)
for i in minRowIndex AI .. maxRowIndex AI repeat
for j in offset + i + 1 .. maxColIndex AI repeat
qsetelt!(AI,i,j, - (((+/[qelt(AI,i,k) * qelt(A,k-offset,j)
for k in i+offset..(j-1)])
exquo qelt(A, j-offset, j))::R))
AI
LowTriBddDenomInv(A, denom) ==
AI := zero(nrows A, nrows A)$M
offset := minColIndex AI - minRowIndex AI
for i in minRowIndex AI .. maxRowIndex AI
for j in minColIndex AI .. maxColIndex AI repeat
qsetelt!(AI,i,j,(denom exquo qelt(A,i,j))::R)
for i in minColIndex AI .. maxColIndex AI repeat
for j in i - offset + 1 .. maxRowIndex AI repeat
qsetelt!(AI,j,i, - (((+/[qelt(A,j,k+offset) * qelt(AI,k,i)
for k in i-offset..(j-1)])
exquo qelt(A, j, j+offset))::R))
AI
)abbrev package IBATOOL IntegralBasisTools
++ Functions common to both integral basis packages
++ Author: Victor Miller, Barry Trager, Clifton Williamson
++ Date Created: 11 April 1990
++ Date Last Updated: 20 September 1994
++ Keywords: integral basis, function field, number field
++ Examples:
++ References:
++ Description:
++ This package contains functions used in the packages
++ FunctionFieldIntegralBasis and NumberFieldIntegralBasis.
IntegralBasisTools(R,UP,F): Exports == Implementation where
R : EuclideanDomain with
squareFree: $ -> Factored $
++ squareFree(x) returns a square-free factorisation of x
UP : UnivariatePolynomialCategory R
F : FramedAlgebra(R,UP)
Mat ==> Matrix R
NNI ==> NonNegativeInteger
Ans ==> Record(basis: Mat, basisDen: R, basisInv:Mat)
Exports ==> with
diagonalProduct: Mat -> R
++ diagonalProduct(m) returns the product of the elements on the
++ diagonal of the matrix m
matrixGcd: (Mat,R,NNI) -> R
++ matrixGcd(mat,sing,n) is \spad{gcd(sing,g)} where \spad{g} is the
++ gcd of the entries of the \spad{n}-by-\spad{n} upper-triangular
++ matrix \spad{mat}.
divideIfCan!: (Matrix R,Matrix R,R,Integer) -> R
++ divideIfCan!(matrix,matrixOut,prime,n) attempts to divide the
++ entries of \spad{matrix} by \spad{prime} and store the result in
++ \spad{matrixOut}. If it is successful, 1 is returned and if not,
++ \spad{prime} is returned. Here both \spad{matrix} and
++ \spad{matrixOut} are \spad{n}-by-\spad{n} upper triangular matrices.
leastPower: (NNI,NNI) -> NNI
++ leastPower(p,n) returns e, where e is the smallest integer
++ such that \spad{p **e >= n}
idealiser: (Mat,Mat) -> Mat
++ idealiser(m1,m2) computes the order of an ideal defined by m1 and m2
idealiser: (Mat,Mat,R) -> Mat
++ idealiser(m1,m2,d) computes the order of an ideal defined by m1 and m2
++ where d is the known part of the denominator
idealiserMatrix: (Mat, Mat) -> Mat
++ idealiserMatrix(m1, m2) returns the matrix representing the linear
++ conditions on the Ring associatied with an ideal defined by m1 and m2.
moduleSum: (Ans,Ans) -> Ans
++ moduleSum(m1,m2) returns the sum of two modules in the framed
++ algebra \spad{F}. Each module \spad{mi} is represented as follows:
++ F is a framed algebra with R-module basis \spad{w1,w2,...,wn} and
++ \spad{mi} is a record \spad{[basis,basisDen,basisInv]}. If
++ \spad{basis} is the matrix \spad{(aij, i = 1..n, j = 1..n)}, then
++ a basis \spad{v1,...,vn} for \spad{mi} is given by
++ \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 \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
Implementation ==> add
import ModularHermitianRowReduction(R)
import TriangularMatrixOperations(R, Vector R, Vector R, Matrix R)
diagonalProduct m ==
ans : R := 1
for i in minRowIndex m .. maxRowIndex m
for j in minColIndex m .. maxColIndex m repeat
ans := ans * qelt(m, i, j)
ans
matrixGcd(mat,sing,n) ==
-- note: 'matrix' is upper triangular;
-- no need to do anything below the diagonal
d := sing
for i in 1..n repeat
for j in i..n repeat
if not zero?(mij := qelt(mat,i,j)) then d := gcd(d,mij)
one? d => return d
d
divideIfCan!(matrix,matrixOut,prime,n) ==
-- note: both 'matrix' and 'matrixOut' will be upper triangular;
-- no need to do anything below the diagonal
for i in 1..n repeat
for j in i..n repeat
(a := (qelt(matrix,i,j) exquo prime)) case "failed" => return prime
qsetelt!(matrixOut,i,j,a :: R)
1
leastPower(p,n) ==
-- efficiency is not an issue here
e : NNI := 1; q := p
while q < n repeat (e := e + 1; q := q * p)
e
idealiserMatrix(ideal,idealinv) ==
-- computes the Order of the ideal
n := rank()$F
bigm := zero(n * n,n)$Mat
mr := minRowIndex bigm; mc := minColIndex bigm
v := basis()$F
for i in 0..n-1 repeat
r := regularRepresentation qelt(v,i + minIndex v)
m := ideal * r * idealinv
for j in 0..n-1 repeat
for k in 0..n-1 repeat
bigm(j * n + k + mr,i + mc) := qelt(m,j + mr,k + mc)
bigm
idealiser(ideal,idealinv) ==
bigm := idealiserMatrix(ideal, idealinv)
transpose squareTop rowEch bigm
idealiser(ideal,idealinv,denom) ==
bigm := (idealiserMatrix(ideal, idealinv) exquo denom)::Mat
transpose squareTop rowEchelon(bigm,denom)
moduleSum(mod1,mod2) ==
rb1 := mod1.basis; rbden1 := mod1.basisDen; rbinv1 := mod1.basisInv
rb2 := mod2.basis; rbden2 := mod2.basisDen; rbinv2 := mod2.basisInv
-- compatibility check: doesn't take much computation time
(not square? rb1) or (not square? rbinv1) or (not square? rb2) _
or (not square? rbinv2) =>
error "moduleSum: matrices must be square"
((n := nrows rb1) ~= (nrows rbinv1)) or (n ~= (nrows rb2)) _
or (n ~= (nrows rbinv2)) =>
error "moduleSum: matrices of imcompatible dimensions"
(zero? rbden1) or (zero? rbden2) =>
error "moduleSum: denominator must be non-zero"
den := lcm(rbden1,rbden2); c1 := den quo rbden1; c2 := den quo rbden2
rb := squareTop rowEchelon(vertConcat(c1 * rb1,c2 * rb2),den)
rbinv := UpTriBddDenomInv(rb,den)
[rb,den,rbinv]
)abbrev package FFINTBAS FunctionFieldIntegralBasis
++ Integral bases for function fields of dimension one
++ Author: Victor Miller
++ Date Created: 9 April 1990
++ Date Last Updated: 20 September 1994
++ Keywords:
++ Examples:
++ References:
++ Description:
++ In this package R is a Euclidean domain and F is a framed algebra
++ over R. The package provides functions to compute the integral
++ closure of R in the quotient field of F. It is assumed that
++ \spad{char(R/P) = char(R)} for any prime P of R. A typical instance of
++ this is when \spad{R = K[x]} and F is a function field over R.
FunctionFieldIntegralBasis(R,UP,F): Exports == Implementation where
R : EuclideanDomain with
squareFree: $ -> Factored $
++ squareFree(x) returns a square-free factorisation of x
UP : UnivariatePolynomialCategory R
F : FramedAlgebra(R,UP)
I ==> Integer
Mat ==> Matrix R
NNI ==> NonNegativeInteger
Exports ==> with
integralBasis : () -> Record(basis: Mat, basisDen: R, basisInv:Mat)
++ \spad{integralBasis()} returns a record
++ \spad{[basis,basisDen,basisInv]} containing information regarding
++ the integral closure of R in the quotient field of F, where
++ F is a framed algebra with R-module basis \spad{w1,w2,...,wn}.
++ If \spad{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 \spad{basis} contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
localIntegralBasis : R -> Record(basis: Mat, basisDen: R, basisInv:Mat)
++ \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 F, where F is a framed algebra with R-module basis
++ \spad{w1,w2,...,wn}.
++ If \spad{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 \spad{basis} contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
Implementation ==> add
import IntegralBasisTools(R, UP, F)
import ModularHermitianRowReduction(R)
import TriangularMatrixOperations(R, Vector R, Vector R, Matrix R)
squaredFactors: R -> R
squaredFactors px ==
*/[(if ffe.exponent > 1 then ffe.factor else 1$R)
for ffe in factors squareFree px]
iIntegralBasis: (Mat,R,R) -> Record(basis: Mat, basisDen: R, basisInv:Mat)
iIntegralBasis(tfm,disc,sing) ==
-- tfm = trace matrix of current order
n := rank()$F; tfm0 := copy tfm; disc0 := disc
rb := scalarMatrix(n, 1); rbinv := scalarMatrix(n, 1)
-- rb = basis matrix of current order
-- rbinv = inverse basis matrix of current order
-- these are wrt the original basis for F
rbden : R := 1; index : R := 1; oldIndex : R := 1
-- rbden = denominator for current basis matrix
-- index = index of original order in current order
not sizeLess?(1, sing) => [rb, rbden, rbinv]
repeat
-- compute the p-radical
idinv := transpose squareTop rowEchelon(tfm, sing)
-- [u1,..,un] are the coordinates of an element of the p-radical
-- iff [u1,..,un] * idinv is in sing * R^n
id := rowEchelon LowTriBddDenomInv(idinv, sing)
-- id = basis matrix of the p-radical
idinv := UpTriBddDenomInv(id, sing)
-- id * idinv = sing * identity
-- no need to check for inseparability in this case
rbinv := idealiser(id * rb, rbinv * idinv, sing * rbden)
index := diagonalProduct rbinv
rb := rowEchelon LowTriBddDenomInv(rbinv, rbden * sing)
g := matrixGcd(rb,sing,n)
if sizeLess?(1,g) then rb := (rb exquo g) :: Mat
rbden := rbden * (sing quo g)
rbinv := UpTriBddDenomInv(rb, rbden)
disc := disc0 quo (index * index)
indexChange := index quo oldIndex; oldIndex := index
sing := gcd(indexChange, squaredFactors disc)
not sizeLess?(1, sing) => return [rb, rbden, rbinv]
tfm := ((rb * tfm0 * transpose rb) exquo (rbden * rbden)) :: Mat
integralBasis() ==
n := rank()$F; p := characteristic$F
(not zero? p) and (n >= p) =>
error "integralBasis: possible wild ramification"
tfm := traceMatrix()$F; disc := determinant tfm
sing := squaredFactors disc -- singularities of relative Spec
iIntegralBasis(tfm,disc,sing)
localIntegralBasis prime ==
n := rank()$F; p := characteristic$F
(not zero? p) and (n >= p) =>
error "integralBasis: possible wild ramification"
tfm := traceMatrix()$F; disc := determinant tfm
(disc exquo (prime * prime)) case "failed" =>
[scalarMatrix(n,1),1,scalarMatrix(n,1)]
iIntegralBasis(tfm,disc,prime)
)abbrev package WFFINTBS WildFunctionFieldIntegralBasis
++ Authors: Victor Miller, Clifton Williamson
++ Date Created: 24 July 1991
++ Date Last Updated: 20 September 1994
++ Basic Operations: integralBasis, localIntegralBasis
++ Related Domains: IntegralBasisTools(R,UP,F),
++ TriangularMatrixOperations(R,Vector R,Vector R,Matrix R)
++ Also See: FunctionFieldIntegralBasis, NumberFieldIntegralBasis
++ AMS Classifications:
++ Keywords: function 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 framed algebra over R. 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.
WildFunctionFieldIntegralBasis(K,R,UP,F): Exports == Implementation where
K : FiniteFieldCategory
--K : Join(Field,Finite)
R : UnivariatePolynomialCategory K
UP : UnivariatePolynomialCategory R
F : FramedAlgebra(R,UP)
I ==> Integer
Mat ==> Matrix R
NNI ==> NonNegativeInteger
SAE ==> SimpleAlgebraicExtension
RResult ==> Record(basis: Mat, basisDen: R, basisInv:Mat)
IResult ==> Record(basis: Mat, basisDen: R, basisInv:Mat,discr: R)
MATSTOR ==> StorageEfficientMatrixOperations
Exports ==> with
integralBasis : () -> RResult
++ \spad{integralBasis()} returns a record
++ \spad{[basis,basisDen,basisInv]} containing information regarding
++ the integral closure of R in the quotient field of F, where
++ F is a framed algebra with R-module basis \spad{w1,w2,...,wn}.
++ If \spad{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 \spad{basis} contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
localIntegralBasis : R -> RResult
++ \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 F, where F is a framed algebra with R-module basis
++ \spad{w1,w2,...,wn}.
++ If \spad{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 \spad{basis} contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
Implementation ==> add
import IntegralBasisTools(R, UP, F)
import ModularHermitianRowReduction(R)
import TriangularMatrixOperations(R, Vector R, Vector R, Matrix R)
import DistinctDegreeFactorize(K,R)
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)$DistinctDegreeFactorize(K,R)
for f in factors(factored) repeat
if f.exponent > 1 then ans := concat(f.factor,ans)
ans
iLocalIntegralBasis: (Vector F,Vector F,Matrix R,Matrix R,R,R) -> IResult
iLocalIntegralBasis(bas,pows,tfm,matrixOut,disc,prime) ==
n := rank()$F; standardBasis := basis()$F
-- 'standardBasis' is the basis for F as a FramedAlgebra;
-- usually this is [1,y,y**2,...,y**(n-1)]
p2 := prime * prime; sae := SAE(K,R,prime)
p := characteristic$F; q := size()$sae
lp := leastPower(q,n)
rb := scalarMatrix(n,1); rbinv := scalarMatrix(n,1)
-- rb = basis matrix of current order
-- rbinv = inverse basis matrix of current order
-- these are wrt the orginal basis for F
rbden : R := 1; index : R := 1; oldIndex : R := 1
-- rbden = denominator for current basis matrix
-- index = index of original order in current order
repeat
-- pows = [(w1 * rbden) ** q,...,(wn * rbden) ** q], where
-- bas = [w1,...,wn] is 'rbden' times the basis for the order B = 'rb'
for i in 1..n repeat
bi : F := 0
for j in 1..n repeat
bi := bi + qelt(rb,i,j) * qelt(standardBasis,j)
qsetelt!(bas,i,bi)
qsetelt!(pows,i,bi ** p)
coor0 := transpose coordinates(pows,bas)
denPow := rbden ** ((p - 1) :: NNI)
(coMat0 := coor0 exquo denPow) case "failed" =>
error "can't happen"
-- the jth column of coMat contains the coordinates of (wj/rbden)**q
-- with respect to the basis [w1/rbden,...,wn/rbden]
coMat := coMat0 :: Matrix R
-- the ith column of 'pPows' contains the coordinates of the pth power
-- of the ith basis element for B/prime.B over 'sae' = R/prime.R
pPows := map(reduce,coMat)$MatrixCategoryFunctions2(R,Vector R,
Vector R,Matrix R,sae,Vector sae,Vector sae,Matrix sae)
-- 'frob' will eventually be the Frobenius matrix for B/prime.B over
-- 'sae' = R/prime.R; at each stage of the loop the ith column will
-- contain the coordinates of p^k-th powers of the ith basis element
frob := copy pPows; tmpMat : Matrix sae := new(n,n,0)
for r in 2..leastPower(p,q) repeat
for i in 1..n repeat for j in 1..n repeat
qsetelt!(tmpMat,i,j,qelt(frob,i,j) ** p)
times!(frob,pPows,tmpMat)$MATSTOR(sae)
frobPow := frob ** lp
-- compute the p-radical
ns := nullSpace frobPow
for i in 1..n repeat for j in 1..n repeat qsetelt!(tfm,i,j,0)
for vec in ns for i in 1.. repeat
for j in 1..n repeat
qsetelt!(tfm,i,j,lift qelt(vec,j))
id := squareTop rowEchelon(tfm,prime)
-- id = basis matrix of the p-radical
idinv := UpTriBddDenomInv(id, prime)
-- id * idinv = prime * identity
-- no need to check for inseparability in this case
rbinv := idealiser(id * rb, rbinv * idinv, prime * rbden)
index := diagonalProduct rbinv
rb := rowEchelon LowTriBddDenomInv(rbinv,rbden * prime)
if divideIfCan!(rb,matrixOut,prime,n) = 1
then rb := matrixOut
else rbden := rbden * prime
rbinv := UpTriBddDenomInv(rb,rbden)
indexChange := index quo oldIndex
oldIndex := index
disc := disc quo (indexChange * indexChange)
(not sizeLess?(1,indexChange)) or ((disc exquo p2) case "failed") =>
return [rb, rbden, rbinv, disc]
integralBasis() ==
traceMat := traceMatrix()$F; n := rank()$F
disc := determinant traceMat -- discriminant of current order
zero? disc => error "integralBasis: polynomial must be separable"
singList := listSquaredFactors disc -- singularities of relative Spec
runningRb := scalarMatrix(n,1); runningRbinv := scalarMatrix(n,1)
-- runningRb = basis matrix of current order
-- runningRbinv = inverse basis matrix of current order
-- these are wrt the original basis for F
runningRbden : R := 1
-- runningRbden = denominator for current basis matrix
empty? singList => [runningRb, runningRbden, runningRbinv]
bas : Vector F := new(n,0); pows : Vector F := new(n,0)
-- storage for basis elements and their powers
tfm : Matrix R := new(n,n,0)
-- 'tfm' will contain the coordinates of a lifting of the kernel
-- of a power of Frobenius
matrixOut : Matrix R := new(n,n,0)
for prime in singList repeat
lb := iLocalIntegralBasis(bas,pows,tfm,matrixOut,disc,prime)
rb := lb.basis; rbinv := lb.basisInv; rbden := lb.basisDen
disc := lb.discr
-- update 'running integral basis' if newly computed
-- local integral basis is non-trivial
if sizeLess?(1,rbden) then
mat := vertConcat(rbden * runningRb,runningRbden * rb)
runningRbden := runningRbden * rbden
runningRb := squareTop rowEchelon(mat,runningRbden)
runningRbinv := UpTriBddDenomInv(runningRb,runningRbden)
[runningRb, runningRbden, runningRbinv]
localIntegralBasis prime ==
traceMat := traceMatrix()$F; n := rank()$F
disc := determinant traceMat -- discriminant of current order
zero? disc => error "localIntegralBasis: polynomial must be separable"
(disc exquo (prime * prime)) case "failed" =>
[scalarMatrix(n,1), 1, scalarMatrix(n,1)]
bas : Vector F := new(n,0); pows : Vector F := new(n,0)
-- storage for basis elements and their powers
tfm : Matrix R := new(n,n,0)
-- 'tfm' will contain the coordinates of a lifting of the kernel
-- of a power of Frobenius
matrixOut : Matrix R := new(n,n,0)
lb := iLocalIntegralBasis(bas,pows,tfm,matrixOut,disc,prime)
[lb.basis, lb.basisDen, lb.basisInv]
)abbrev package NFINTBAS NumberFieldIntegralBasis
++ Author: Victor Miller, Clifton Williamson
++ Date Created: 9 April 1990
++ Date Last Updated: 20 September 1994
++ Basic Operations: discriminant, integralBasis
++ Related Domains: IntegralBasisTools, TriangularMatrixOperations
++ Also See: FunctionFieldIntegralBasis, WildFunctionFieldIntegralBasis
++ AMS Classifications:
++ Keywords: number field, integral basis, discriminant
++ Examples:
++ References:
++ Description:
++ In this package F is a framed algebra over the integers (typically
++ \spad{F = Z[a]} for some algebraic integer a). The package provides
++ functions to compute the integral closure of Z in the quotient
++ quotient field of F.
NumberFieldIntegralBasis(UP,F): Exports == Implementation where
UP : UnivariatePolynomialCategory Integer
F : FramedAlgebra(Integer,UP)
FR ==> Factored Integer
I ==> Integer
Mat ==> Matrix I
NNI ==> NonNegativeInteger
Ans ==> Record(basis: Mat, basisDen: I, basisInv:Mat,discr: I)
Exports ==> with
discriminant: () -> Integer
++ \spad{discriminant()} returns the discriminant of the integral
++ closure of Z in the quotient field of the framed algebra F.
integralBasis : () -> Record(basis: Mat, basisDen: I, basisInv:Mat)
++ \spad{integralBasis()} returns a record
++ \spad{[basis,basisDen,basisInv]}
++ containing information regarding the integral closure of Z in the
++ quotient field of F, where F is a framed algebra with Z-module
++ basis \spad{w1,w2,...,wn}.
++ If \spad{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 \spad{basis} contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
localIntegralBasis : I -> Record(basis: Mat, basisDen: I, basisInv:Mat)
++ \spad{integralBasis(p)} returns a record
++ \spad{[basis,basisDen,basisInv]} containing information regarding
++ the local integral closure of Z at the prime \spad{p} in the quotient
++ field of F, where F is a framed algebra with Z-module basis
++ \spad{w1,w2,...,wn}.
++ If \spad{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 \spad{basis} contains the coordinates of the
++ \spad{i}th basis vector. Similarly, the \spad{i}th row of the
++ matrix \spad{basisInv} contains the coordinates of \spad{wi} with
++ respect to the basis \spad{v1,...,vn}: if \spad{basisInv} is the
++ matrix \spad{(bij, i = 1..n, j = 1..n)}, then
++ \spad{wi = sum(bij * vj, j = 1..n)}.
Implementation ==> add
import IntegralBasisTools(I, UP, F)
import ModularHermitianRowReduction(I)
import TriangularMatrixOperations(I, Vector I, Vector I, Matrix I)
frobMatrix : (Mat,Mat,I,NNI) -> Mat
wildPrimes : (FR,I) -> List I
tameProduct : (FR,I) -> I
iTameLocalIntegralBasis : (Mat,I,I) -> Ans
iWildLocalIntegralBasis : (Mat,I,I) -> Ans
frobMatrix(rb,rbinv,rbden,p) ==
n := rank()$F; b := basis()$F
v : Vector F := new(n,0)
for i in minIndex(v)..maxIndex(v)
for ii in minRowIndex(rb)..maxRowIndex(rb) repeat
a : F := 0
for j in minIndex(b)..maxIndex(b)
for jj in minColIndex(rb)..maxColIndex(rb) repeat
a := a + qelt(rb,ii,jj) * qelt(b,j)
qsetelt!(v,i,a**p)
mat := transpose coordinates v
((transpose(rbinv) * mat) exquo (rbden ** p)) :: Mat
wildPrimes(factoredDisc,n) ==
-- returns a list of the primes <=n which divide factoredDisc to a
-- power greater than 1
ans : List I := empty()
for f in factors(factoredDisc) repeat
if f.exponent > 1 and f.factor <= n then ans := concat(f.factor,ans)
ans
tameProduct(factoredDisc,n) ==
-- returns the product of the primes > n which divide factoredDisc
-- to a power greater than 1
ans : I := 1
for f in factors(factoredDisc) repeat
if f.exponent > 1 and f.factor > n then ans := f.factor * ans
ans
integralBasis() ==
traceMat := traceMatrix()$F; n := rank()$F
disc := determinant traceMat -- discriminant of current order
disc0 := disc -- this is disc(F)
factoredDisc := factor(disc0)$IntegerFactorizationPackage(Integer)
wilds := wildPrimes(factoredDisc,n)
sing := tameProduct(factoredDisc,n)
runningRb := scalarMatrix(n, 1); runningRbinv := scalarMatrix(n, 1)
-- runningRb = basis matrix of current order
-- runningRbinv = inverse basis matrix of current order
-- these are wrt the original basis for F
runningRbden : I := 1
-- runningRbden = denominator for current basis matrix
one? sing and empty? wilds => [runningRb, runningRbden, runningRbinv]
-- id = basis matrix of the ideal (p-radical) wrt current basis
matrixOut : Mat := scalarMatrix(n,0)
for p in wilds repeat
lb := iWildLocalIntegralBasis(matrixOut,disc,p)
rb := lb.basis; rbinv := lb.basisInv; rbden := lb.basisDen
disc := lb.discr
-- update 'running integral basis' if newly computed
-- local integral basis is non-trivial
if sizeLess?(1,rbden) then
mat := vertConcat(rbden * runningRb,runningRbden * rb)
runningRbden := runningRbden * rbden
runningRb := squareTop rowEchelon(mat,runningRbden)
runningRbinv := UpTriBddDenomInv(runningRb,runningRbden)
lb := iTameLocalIntegralBasis(traceMat,disc,sing)
rb := lb.basis; rbinv := lb.basisInv; rbden := lb.basisDen
disc := lb.discr
-- update 'running integral basis' if newly computed
-- local integral basis is non-trivial
if sizeLess?(1,rbden) then
mat := vertConcat(rbden * runningRb,runningRbden * rb)
runningRbden := runningRbden * rbden
runningRb := squareTop rowEchelon(mat,runningRbden)
runningRbinv := UpTriBddDenomInv(runningRb,runningRbden)
[runningRb,runningRbden,runningRbinv]
localIntegralBasis p ==
traceMat := traceMatrix()$F; n := rank()$F
disc := determinant traceMat -- discriminant of current order
(disc exquo (p*p)) case "failed" =>
[scalarMatrix(n, 1), 1, scalarMatrix(n, 1)]
lb :=
p > rank()$F =>
iTameLocalIntegralBasis(traceMat,disc,p)
iWildLocalIntegralBasis(scalarMatrix(n,0),disc,p)
[lb.basis,lb.basisDen,lb.basisInv]
iTameLocalIntegralBasis(traceMat,disc,sing) ==
n := rank()$F; disc0 := disc
rb := scalarMatrix(n, 1); rbinv := scalarMatrix(n, 1)
-- rb = basis matrix of current order
-- rbinv = inverse basis matrix of current order
-- these are wrt the original basis for F
rbden : I := 1; index : I := 1; oldIndex : I := 1
-- rbden = denominator for current basis matrix
-- id = basis matrix of the ideal (p-radical) wrt current basis
tfm := traceMat
repeat
-- compute the p-radical = p-trace-radical
idinv := transpose squareTop rowEchelon(tfm,sing)
-- [u1,..,un] are the coordinates of an element of the p-radical
-- iff [u1,..,un] * idinv is in p * Z^n
id := rowEchelon LowTriBddDenomInv(idinv, sing)
-- id = basis matrix of the p-radical
idinv := UpTriBddDenomInv(id, sing)
-- id * idinv = sing * identity
-- no need to check for inseparability in this case
rbinv := idealiser(id * rb, rbinv * idinv, sing * rbden)
index := diagonalProduct rbinv
rb := rowEchelon LowTriBddDenomInv(rbinv, sing * rbden)
g := matrixGcd(rb,sing,n)
if sizeLess?(1,g) then rb := (rb exquo g) :: Mat
rbden := rbden * (sing quo g)
rbinv := UpTriBddDenomInv(rb, rbden)
disc := disc0 quo (index * index)
indexChange := index quo oldIndex; oldIndex := index
one? indexChange => return [rb, rbden, rbinv, disc]
tfm := ((rb * traceMat * transpose rb) exquo (rbden * rbden)) :: Mat
iWildLocalIntegralBasis(matrixOut,disc,p) ==
n := rank()$F; disc0 := disc
rb := scalarMatrix(n, 1); rbinv := scalarMatrix(n, 1)
-- rb = basis matrix of current order
-- rbinv = inverse basis matrix of current order
-- these are wrt the original basis for F
rbden : I := 1; index : I := 1; oldIndex : I := 1
-- rbden = denominator for current basis matrix
-- id = basis matrix of the ideal (p-radical) wrt current basis
p2 := p * p; lp := leastPower(p::NNI,n)
repeat
tfm := frobMatrix(rb,rbinv,rbden,p::NNI) ** lp
-- compute Rp = p-radical
idinv := transpose squareTop rowEchelon(tfm, p)
-- [u1,..,un] are the coordinates of an element of Rp
-- iff [u1,..,un] * idinv is in p * Z^n
id := rowEchelon LowTriBddDenomInv(idinv,p)
-- id = basis matrix of the p-radical
idinv := UpTriBddDenomInv(id,p)
-- id * idinv = p * identity
-- no need to check for inseparability in this case
rbinv := idealiser(id * rb, rbinv * idinv, p * rbden)
index := diagonalProduct rbinv
rb := rowEchelon LowTriBddDenomInv(rbinv, p * rbden)
if divideIfCan!(rb,matrixOut,p,n) = 1
then rb := matrixOut
else rbden := p * rbden
rbinv := UpTriBddDenomInv(rb, rbden)
indexChange := index quo oldIndex; oldIndex := index
disc := disc quo (indexChange * indexChange)
one? indexChange or gcd(p2,disc) ~= p2 =>
return [rb, rbden, rbinv, disc]
discriminant() ==
disc := determinant traceMatrix()$F
intBas := integralBasis()
rb := intBas.basis; rbden := intBas.basisDen
index := ((rbden ** rank()$F) exquo (determinant rb)) :: Integer
(disc exquo (index * index)) :: Integer
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