=============================
new functions and new arguments in KASH 2.2
=============================
ColorString -> computes a string that can be used for
colorful printing in kash programs
Bell -> sounds the terminal bell
Colors -> added possibility to change defaults
_ColorTable -> here are the defaults, have a look
IsRecType -> IsRecType(r, string) is true iff
r is a record, r.Type is existent and
r.Type = string
RecDump -> Outputs a record and all entries,
circumvents the operations.Print
entry
MatHermiteRowUpper -> stupid, but new
MatHermiteRowUpperTrans -> sbn
AbelianFieldToRCF -> starting with an abelian field, we'll
reconstruct an RayClassGroupToAbelianGroup
AbelianGroupEltRandom -> random element of a finite abelian group
AbelianGroupEnumInit -> environment for the enumeration of all
AbelianGroupEnumNext group elements
AbelianGroupHomCreateId -> trivial homomorphism
AbelianRayClassGroupAutoCreate -> starting with an automorphism of ideals,
we'll produce an automorphism of groups
FindMaximalCentralField -> find maximal quotient of
RayClassGroupToAbelianGroup that will
define an central extension
FindQuotientOfShapeEnumInit -> environment to find subgroups/ quotients
FindQuotientOfShapeEnumNext of a certain shape
IdealPrimeCountInit -> environment to enumerate prime ideals
IdealPrimeCountNext or their norms
MatIndex -> s.th. like the determinant of rectangular
matrices.
RayClassFieldIsCentral -> tests whether a RayClassField is going to
be a central extension
RayClassFieldIsNormal -> same for normal
RayClassFieldSplittingField -> computes the minimal splitting field
resp. the data neccessary to get a defining
equation via RayClassField
(for more or less normal base fields!!)
RayClassGroupToAbelianGroup -> changed: ideal, inf, MATRIX,
Group name added
RayClassField -> Group as parameter added,
Group, deg
Group, Matrix
RayConductor -> Group as parameter
EltPowerProduct
MatSmith -> FFx allowed
MatSmithTrans
#
Galois, GaloisT, OrderGalois -> optional parameter "usc" is removed,
because it is done automatically in
certain cases (e.g. for primitive groups)
The result is unconditional in this case,
too.
# the main new Alff functions:
AlffCanonicalDivisor -> compute a canonical divisor
AlffClassGroup -> compute the group structure of the
group of divisor classes of degree zero
of a global function field
AlffClassGroupGens -> same, but additionally return generators
of the divisor class group
AlffClassGroupPRank -> compute the p-rank of the class group,
p = characteristic of the global function
field
AlffClassGroupGenBound -> compute a degree bound for the prime
divisors which generate the divisor class
group together with the supp of a divisor
of degree one of a global function field
AlffClassGroupGenBoundStrong -> same, but better by inspecting the gff
AlffClassNumberApprox -> approximate the class number
AlffClassNumberApproxBound -> compute a certain bound for class number
approximations
AlffDivisorClassRep -> compute the class representation
of a divisor in the generators of the
divisor class group of a global
function field
AlffDivisorLargeLDim -> compute the dimension of a divisor
of large degree or with large exponents
AlffDivisorLargeLBasisShort -> compute the basis of the Riemann-Roch
space of a divisor of large degree or
with large exponents
AlffDivisorReduction -> compute a divisor reduction
AlffDivisorsSmoothNum -> compute the number of smooth divisors
AlffDiffCartier -> compute the Cartier operator
AlffDiffCartierMatrix -> return the representation matrix
of the Cartier-operator on a basis
of holomorphic differentials
AlffDifferent -> return the different of F / k(x)
AlffDiffSpace -> return a space of differentials >= a divisor
AlffDiffFirstKind -> return a basis of the differentials of first
kind (= holomorphic differentials)
AlffDifferentiation -> compute higher differentiations a la Hasse
AlffDiff* -> more functions for differentials
AlffEltEval -> eval an algebraic function at a place
AlffEltLift -> compute an algebraic function which
takes a prescribed value at a place
AlffEltPthRoot, QfePthRoot -> take a p-th root in a global function
field of characteristic p
AlffGapNumbers -> return the gap numbers of a place of degree
one
AlffHasseWittInvariant -> compute the Hasse-Witt invariant of a global
function field
AlffIdealClassGroupUnitsInfty -> compute units and ideal class group of
the finite maximal order
AlffLinearSeries* -> 3 functions for the enumeration
of a linear series in a global function
field
AlffLPoly -> compute the L-polynomial
AlffLPolyLift -> lift the L-polynomial to constant field
extensions
AlffLPolyRed -> compute the L-polynomial mod p via
the Cartier-operator (after Manin, R"uck)
AlffPlaceRandom -> return a random place of given degree
for a global function field
AlffPlacesNonSpecial -> return a system of non-special places
for a global function field
AlffResidueField -> compute the residue field of a place
AlffSUnits -> compute S-units for a set of places
of a global function field
AlffWeierstrassPlaces -> compute Weierstrass places
PolyPrimeList -> return a list of prime polynomials
of given degree over a finite field
=============================
Renamed functions in KASH 2.2
=============================
=============================
new functions and new arguments in KASH 2.1
=============================
GaloisBlocks -> second optional parameter 'true' possible:
Is the second parameter used, this
function computes the Galois groups
of the subfields and excludes transitive
groups with other Galois groups of the
subfields from the list of possible Galois
groups.
=============================
Renamed functions in KASH 2.1
=============================
=============================
new functions and new arguments in KASH 2.0
=============================
package for AbelianGroups -> basic functions for Abelian groups
RayConductorTest -> support all an aditionally argument
RayConductor to specify a subgroup. The fxn's
RayDiscSig will operate on the factor group.
ZIdealCreate -> generates an ideal in Z
RayClassGroup, RayResidueRing -> work for ZIdeals
RayClassFieldAuto -> computes automorphisms of RayClassFields
RayClassFieldArtin -> the Artin map for RayClassFields
RayClassField -> supports subgroups
Alffs -> the possible constant fields are
now extended to finite fields,
rational numbers, number fields.
Handling of places, divisors,
Riemann-Roch spaces,
maximal orders (integral closures)
are now possible over these
constant fields.
AlffInit -> now gets the constant field as parameter,
variable names can be specified
optionally (standard is T and y).
AlffVarT -> Returns the variable 'T' of the
defining equation of an alff.
AlffVarY -> Returns the variable 'y' of the
defining equation of an alff.
StarkUnitsHilbert -> Determines a defining equation for the
Hilbert Class Field.
StarkUnitsPolynom -> Determines the minimal polynomial of the
Stark-Unit used to compute the Hilbert
Class Field.
StarkUnitsRealPolynom -> Determines a real approximation of the
minimal polynomial of a primitive element
generating the Hilbert class field of a
totally real algebraic number field.
EltCharPoly -> Characteristic polynomial of an algebraic
element over a subfield
CharPoly -> Characteristic polynomial of an algebraic
function field element or a matrix
Den -> generic function, returns the denominator
for the following arguments:
rational, algebraic element, algebraic
function field order element, quotient
field or polynomial, ideal
Num -> generic function, returns the nominator
for the following arguments:
rational, algebraic element, algebraic
function field order element, quotient
field or polynomial, ideal
GaloisSymb -> Unconditional Galois group computation up
to degree 7 for polynomials in Q, Q(x)
and simple relative orders. The name of
the Galois group is returned.
GaloisSymbT -> Same as GaloisSymb, but the number of the
Galois group is returned
Galois, GaloisT, OrderGalois -> now supported up to degree 15
GaloisMSumPol -> Let f be a monic polynomial of degree n
in Q, Q(x) or a simple relative order.This
function computes a primitive polynomial
of degree binmoial(n,m). The roots of
GaloisMSumPol are the sums of m distinct
roots of f.
GaloisMSetPol -> Let f be a monic polynomial of degree n
in Q, Q(x) or a simple relative order.This
function computes a primitive polynomial
of degree binmoial(n,m). The roots of
GaloisMSetPol are the products of m
distinct roots of f.
GaloisTwoSequencePol -> Let f be a monic polynomial of degree n
in Q, Q(x) or a simple relative order.This
function computes a primitive polynomial
of degree n*(n-1). The roots of
GaloisTwoSequencePol are the products of
of the form xi+2xj, where xi and xj are
distinct roots of f.
Poly*, Qf* -> the handling of univariate polynomials
and fractions of such is much
improved resp. generalized.
SPrint -> Creates a string instead of printing on
the screen
SScan -> Reads kash objects out of a string
=============================
Renamed functions in KASH 2.0
=============================
Alff* -> All Gff* functions are renamed as
Alff* functions.
AlffInit(p, q) -> AlffInit(FF(p, q)). AlffInit()
gets the constant field as parameter.
=============================
Renamed functions in KASH 1.9
=============================
IntGcdEx -> IntXGcd !! list is now returned
FFEFF -> FFEltFF
=============================
new arguments in KASH 1.9
=============================
OrderMaximal -> The first argument can now be a list
of arguments defining an order
just like in the function ORDER.
After that you can specify up to
four strings with control devices:
Round2, R2: using the round2-algo
Round4, R4: using the round4-algo
RD: using reduced discriminant
NoRD: avoid reduced discriminant
Split: using algebra-splitting
NoSplit: avoid algebra-splitting
Dedekind: using dedekind-test
NoDedekind: avoid dedekind-test
But only one of these devices in each
string.
old new
OrderMaximal(o,"rd Round2") -> OrderMaximal(o,"Round2","RD");
OrderMaximal(o,"rd Round4") -> OrderMaximal(o,"Round4","RD");
For a detailed description of the new features see the
KASH-Reference Manual or type ?OrderMaximal in a KASH-session.
OrderPMaximal -> You can use the same control devices
as above except of "RD" and "NoRD".
For a detailed description of the new features see the
KASH-Reference Manual or type ?OrderPMaximal in a KASH-session.
MatOrthogonal() -> returns now list of orth. Matrix
and transformation matrix
=============================
Renamed functions in KASH 1.8
=============================
CycloField -> OrderCyclotomic
CycloUnits -> OrderCyclotomicUnits
EltIdealResidueRingRep -> EltRayResidueRingRep
EltMinPolyAbs(a) -> EltMinPoly(a,Zx)
EltMinPolyRel(Px,a) -> EltMinPoly(a,Px)
EltNormAbs(a) -> EltNorm(a,Z)
EltRepMatAbs(a) -> EltRepMAt(a,Z)
EltRepMatRel(a,o) -> EltRepMat(a,o)
EltTraceAbs(a) -> EltTrace(a,Z)
IdealIsPrincipalByClassGroup(id) -> IdealIsPrincipal(id,"classgroup")
IdealResidueRing -> RayResidueRing
IdealResidueRingCyclicFactors -> RayResidueRingCyclicFactors
IdealResidueRingRepToElt -> RayResidueRingRepToElt
OrderClassGroupReg(o) -> OrderReg(o,"classgroup")
OrderClassGroupSUnits -> OrderSUnits
OrderClassGroupUnits(o) -> OrderUnits(o,"classgroup")
MatMinPoly(Zx,M) -> MatMinPoly(M)
MatCharPoly(Zx,M) -> MatCharPoly(M)