- Classification is based on the fact that each reflexive polytope is a
subpolytope on a sublattice of a Newton polytope corresponding to a (minimal)
combined weight systems (CWS) [math.AG/0001106].
The enumeration of all (nonreflexive) subpolytopes would be quite inefficient. The program class.x therefore uses a subtle strategy which eliminates facets with distance>1 recursively by either dropping points or going to a sublattice. We call subpolytopes on the original lattice h (honest) and those found on sublattices sl. Data are stored (in RAM and on the hard disk) in a compressed binary format and [except for ".aux"-files] the h-polytopes should always contain all their h-subpolytopes, while the sl-lists only constitute a generating set that may depend on the path of the calculation (which, in turn, may depend on parameter settings). In order to find all subpolytopes on all sublattices the sl-polytopes have to be extracted and fed back into further runs of class.x until all sl-polytopes are gone (see below). Data on the hard disk are either in single BIN-files or (when the files become too large) converted to data-bases (several files zzdb.info zzdb.vn1 ... zzdb.vnm zzdb.sl with date split according to numbers of vertices). - class.x has a number of functionalities (controlled by options, see class.x -h) for combining, subtracting and manipulating binary data. This is now illustrated by the computation of all reflexive subpolytopes on all possible sublattices for the Newton polytope of the degree 24 polynomial w.r.t. the weights 3 3 4 4 10 (the smallest maximal reflexive polytope in 4 dimensions). Input is displayed in bold face:
$ echo "24 3 3 4 4 10" | class.x -f -po zbin # 3 minutes on 3GHz P4
...
800kR-1280 11MB 2461kIP 968kNF-12k 11_46 v17r17 f28r27 85b24 177s 177u 66n
24 3 3 4 4 10 R=798878 +1280sl hit=0 IP=2461059 NF=968137 (12612)
Writing zbin: 798878+1280sl 1181m+14s 9846494b ... # POLY_Dmax=4
Writing zbin: 798878+1308sl 1181m+14s 9847010b ... # POLY_Dmax=5
Writing zbin: 798878+1658sl 1181m+14s 9851469b ... # POLY_Dmax=6
...
Due to dimension-dependent optimizations in Find_Equations()
the path-dependent number of sl-polytopes depends on the setting
of POLY_Dmax at compilation time.
For extracting the sl-polytopes we need to convert the BIN-file:
$ class.x -pi zbin -do zzdb
Read zbin (798878poly +1280sl 9846494b) write zzdb.* (15 files) done (0s)
(for BIN-files only the h-polytopes are extracted by default).
The following has to be iterated until no sl-polytopes are left:
$ class.x -b -di zzdb |poly.x -fp| class.x -f -pi zbin -po zbinS
...
Writing zbinS: 4457+0sl 0m+0s 46913b u18 done: 0s
...
$ class.x -pi zbin -pa zbinS -po zbinHS
...
Writing zbinHS: 803335+0sl 1181m+14s 9877364b
$ class.x -b -pi zbinHS | class.x -f -pi zbinHS -sv -po zbinQ
...
Writing zbinQ: 9+0sl 0m+0s 75b
...
$ class.x -b -pi zbinQ |poly.x -fp| class.x -f -pi zbinHS -po zbinQS
...
Writing zbinQS: 9+0sl 0m+0s 75b ... # nothing new
$ class.x -b -pi zbinQ | class.x -f -pi zbinHS -sv -po zbinQQ
...
Writing zbinQQ: 4+0sl 0m+0s 31b ...
$ class.x -pi zbinQQ -ps zbinQ -po zbinQ2
...
Writing zbinQ2: 0+0sl 0m+0s 0b # nothing new / iteration finished
$ class.x -pi zbinHS -pa zbinQ -po zbinHSQ
...
Writing zbinHSQ: 803344+0sl 1181m+14s 9877439b # Final result.
Alternatively we may base the calculation on the data base zzdb
and first extract the q-polytopes (q for quotient, i.e. reflexive
polytopes on coarser lattice). The following procedure avoids
multiple updates of the h-polytopes (e.g. in case of large data):
$ class.x -b -di zzdb |poly.x -fp| class.x -f -di zzdb -po zbinS
...
Writing zbinS: 4457+0sl 0m+0s 46913b ...
$ class.x -di zzdb -sv -po zbinQ
...
Writing zbinQ: 805+0sl 0m+0s 7532b ...
$ class.x -pi zbinS -pa zbinQ -po zbinSQ
...
Writing zbinSQ: 4466+0sl 0m+0s 46988b
Check for termination of the iteration (sl is already 0):
$ class.x -b -pi zbinSQ |poly.x -fp| class.x -f -di zzdb -po zbinS2
...
Writing zbinS2: 4466+0sl 0m+0s 46988b ... # nothing new (zbinS2=zbinSQ)
$ class.x -b -pi zbinSQ | class.x -f -di zzdb -sv -po zbinQ2
...
Writing zbinQ2: 132+0sl 0m+0s 1232b ...
$ class.x -pi zbinQ2 -ps zbinSQ -po zbinQ2new
...
Writing zbinQ2new: 0+0sl 0m+0s 0b # nothing new / interation finished
$ class.x -di zzdb -pa zbinSQ
...
Writing zzdb: 803344+0sl 1181m+14s 9877439b # Same final result
- Bin-file subtraction is defined as follows: Let zb.1=(h1,s1) and zb.2=(h2,s2) denote the sets of h and sl polytopes on the binary files zb.1 and zb.2. Define the disjoint intersections A=h1&h2, B=h1&s2, C=s1&h2, D=s1&s2. Then the union of zb.1 and zb.2 is (H=h1+h2,S=s1+s2-H) with disjoint unions H=A+H1+H2, S=S1+S2+D for H1=h1-A, H2=H-A, S1=s1-C-D and S2=s2-B-D (note that subsets B,C of s1,s2 cannot be removed from H1,H2)
$ class.x -pi zb.1 -ps zb.2 -po zb.1n # 1n=1-2=(H1=h1-A,S1=s1-C-D)
$ class.x -pi zb.2 -ps zb.1 -po zb.2n # 2n=2-1=(H2=h2-A,S2=s2-B-D)
$ class.x -pi zb.1 -ps zb.1n -po zb.1i # 1i=1-1n=(A,C+D)
$ class.x -pi zb.2 -ps zb.2n -po zb.2i # 2i=2-2n=(A,B+D)
$ class.x -pi zb.1i -ps zb.2n -po zb.AD1 # (A,D)
$ class.x -pi zb.2i -ps zb.1n -po zb.AD2 # (A,D)
$ class.x -pi zb.1i -ps zb.2i -po zb.0C # (0,C)
$ class.x -pi zb.2i -ps zb.1i -po zb.0B # (0,B)
$ class.x -pi zb.1 -pa zb.2 -po zb.12 # (H=H1+H2+A, S=S1+S2+D)
Data base operations are similarly defined: After conversion
from bin to db (direct back-conversion is not implemented):
$ class.x -pi zb.2 -do zzdb2; class.x -pi zb.1 -ds zzdb2 -po zb.1n
$ class.x -di zzdb2 -pa zb.1 -do zzdb.12