Schemes [HB 87]

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.