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A number of new features have been added by Al Kasprzyk to his polytopes
package.
New Features:
- A native version of PALP normal form has been implemented, and is available
via
the intrinsics PALPNormalForm(P) and NormalForm(P). This is the first
implementation of the famous algorithm by Max Kreuzer and Harald Skarke outside of
PALP. An alternative algorithm has also been developed which is significantly faster
when the polytope P
has symmetries.
- AffineNormalForm(P) can be used to obtain the affine normal form of the
maximum dimensional lattice polytope P
.
- Minkowski decompositions of polytopes in arbitrary dimensions can now be computed.
This is based on a result of Klaus Altmann.
- The Newton polytope of a rational function f
(regarded as a Laurent polynomial)
can be calculated via NewtonPolytope(f).
- The new PointProcess(P) and PointProcess(C) intrinsics can be used
to iterate over the points in a polytope P
and a pointed cone C
.
- Given a primitive form v
(i.e. a point in the lattice dual to L
),
ChangeBasis(v) returns a change of basis of L
such that the kernel
of v
is mapped to the standard codimension one lattice.
- A database of all (2
-dimensional) facets of the canonical Fano 3
-dimensional
polytopes has been added. These polygons, which can be accessed via
PolytopeCanonicalFanoDim3Facet, are defined up to affine equivalence.
- A database of small polygons has been implemented. Once installed, this can be
accessed via PolytopeSmallPolygon.
- Reverse database look-up has been implemented in three-dimensions. Given a
polytope P
, the intrinsic DatabaseID(P) will return a sequence of
matching entries in the databases of three-dimensional polytopes. In the
case when P
is a canonical Fano polytope, the latest version of the
canonical3 database must be installed in order to access reverse
look-up data.
Changes:
- The intrinsic IsIntegrallyClosed(P) can be used to determine whether
a polytope P
is integrally closed (i.e. every lattice point in kP
can
be written as the sum of k
lattice points in P
, for all
k
Z
).
The algorithm has been significantly improved, leading to much earlier
detection of a negative answer.
- New special cases have been added to point enumeration for a polytope P
,
significantly improving the time taken under certain extreme choices of
basis (which, unfortunately, includes some presentations of reflexive
simplices that have become standard).
Next: Finite Geometry
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Previous: Geometry