Algebraic Function Fields [HB 57]

Removals and Changes:

• The type of the vector space returned by RiemannRochSpace, Module and Relations has changed.

• The order of the arguments to Module and Relations has changed.

New Features:

• FunctionField(g) where g is a bivariate polynomial.

• For all algebraic function fields RationalExtensionRepresentation, ExactConstantField and SeparatingElement have been added.

• The following new functions apply to function field and/or order elements: IsSeparating, Root, IsConstant, IntegralSplit, Expand, RationalFunction, ProductRepresentation and Minimum. Random elements of global function fields and their orders are now obtainable.

• The following new functions apply to divisors and/or places: Gcd, Lcm, IsCanonical, ShortBasis, ComplementaryDivisor, NumberOfSmoothDivisors, Roots, RamificationDivisor, GapNumbers, WronskianOrders, WeierstrassPlaces and IsWeierstrassPlace. The function Divisor has a new signature that takes two ideals belonging to maximal orders of a function field.

• An algorithm for computing the divisor class group of a global function field has been implemented. As a consequence the following functions relating to the class group may be applied to a global function field: ClassGroup, ClassGroupAbelianInvariants, ClassNumber, PrincipalDivisorMap, ClassGroupExactSequence, GlobalUnitGroup, IsGlobalUnit, SClassGroupAbelianInvariants, IsGlobalUnitWithPreimage, SClassNumber, IsSUnitWithPreimage, IsSPrincipal, SUnitGroup, SClassGroupExactSequence, IsSUnit, SPrincipalDivisorMap, SRegulator, ClassGroupPRank, and HasseWittInvariant.

• The following functions relating to the class group may be applied to maximal finite orders and their ideals in a global function field: ClassGroup, ClassGroupAbelianInvariants, ClassNumber, ClassGroupExactSequence, PrincipalIdealMap, IsPrincipal UnitGroup, IsUnitWithPreimage.

• The function WeilRestriction is available for elliptic function fields.

• Machinery for working with differentials (DiffFunElt) is now available. In addition to basic arithmetic the most relevant functions are DifferentialSpace, DifferentialBasis, Differentiation, Differential, Divisor, SpaceOfDifferentialsFirstKind, BasisOfDifferentialsFirstKind, SpaceOfHolomorphicDifferentials, BasisOfHolomorphicDifferentials, Residue, Valuation, IsCanonical, FunctionField, IsExact, Cartier, and CartierRepresentation.

• The modules returned by RiemannRochSpace, SpaceOfHolomorphicDifferentials, Relations, Module and DifferentialSpace, come with embeddings into a function field or space of differentials of a functions field. The intersection, sum etc. of such modules is possible as well as coercion between the field or space and the module.

• Factorization of polynomials (univariate and multivariate) over function fields and their orders can now be achieved.