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Removals and Changes:
- Improved checking for inverses when the domain is not an ambient.
- AlternateDefiningPolynomials has been renamed to
AllDefiningPolynomials which includes the original definition of
the map.
New Features:
- More checks for well-definedness when multiple sets of defining equations
are given.
- Multiple definitions for the inverse of a map can be given. Such definitions
can be retrieved using AllInverseDefiningPolynomials.
- Inverses of projective closure maps are set on creation.
- The function Difference will remove from a scheme the intersection
of that scheme with a second argument (taking into account multiplicities)
and return the closure of this.
- The functions IntegralSplit, Numerator and Denominator
have been provided for working with function field elements and associated
schemes, as well as AlgebraicFunction and Restriction.
- Projective maps can be created using function field elements by the function
ProjectiveMap.
- A test for the birationality of scheme maps is provided by
IsInvertible. A birational inverse is returned if it exists.
- GenericPoint returns a point in the pointset of the function
field of the scheme.
- Compositions of scheme maps can be multiplied out fully (including multiple
definitions) using Expand. A map between schemes can be Extended
to be defined on the whole domain. For a map with multiple definitions
Prune will remove any unnecessary definitions.
- It is now possible to test whether a pointset over a local field is empty
using IsEmpty. Further one can use IsLocallySolvable to test
whether a scheme over a number field has a local point and LiftPoint
to increase the precision of a point.
- One can check whether a function field can be computed for a scheme
using HasFunctionField.
Bug Fixes:
- Greater strictness on base rings for which Gröbner basis algorithms
are not available. One can no longer retrieve the ideal of a scheme over
a ring without these algorithms or do anything
which requires such information.
Next: Algebraic Curves [HB 88]
Up: Algebraic Geometry
Previous: Algebraic Geometry