General aspects of using PALP

On this page we discuss aspects of PALP that are common to most or all of its applications. The first step is to download the package from this website and follow the compilation instructions given there, which should result in the existence of a directory `palp' containing the program as well as the executable files.

Polytope Input

The majority of applications requires input in the form of a list of polytopes. There are essentially two ways of entering the data of a polytope. Matrix input starts with a line containing two numbers n_lines and n_columns (which may be followed by text which is simply ignored by the program) and proceeds with a matrix with the corresponding numbers of lines and columns. PALP requires $n_{\rm lines} \not=n_{\rm columns}$ and interprets the smaller of the two numbers as the dimension of the polytope and the other one as the number of polytope points entered as lines or columns of input.

palp$ poly.x
Degrees and weights  `d1 w11 w12 ... d2 w21 w22 ...'
  or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
3 2  This text is ignored by PALP
Type the 6 coordinates as #pts=3 lines with dim=2 columns:
2 0
0 2
0 0
M:6 3 F:3
Degrees and weights  `d1 w11 w12 ... d2 w21 w22 ...'
  or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
2 3 The same example with transposed input
Type the 6 coordinates as dim=2 lines with #pts=3 columns:
2 0 0 
0 2 0
M:6 3 F:3

In both cases the input specifies the polygon (2-polytope) that is the convex hull of the 3 points {(2,0),(0,2),(0,0)} in $M=\mathbb{Z}^2$. The output just means that this polygon has 6 lattice points, 3 vertices and 3 facets (here, edges). The possibility of ignored text in the input is useful because PALP's output can often be used as input for further applications; thereby extra information can be displayed without destroying the permissible format.

For applications in the context of toric geometry one should be aware of the fact that there are two relevant, mutually dual lattices M and N whose toric interpretations are quite different. By default PALP interprets the input polytope as $\Delta\subset M_\mathbb{R}$. Note that PALP refers to this polytope as $P$; in this paper we shall use both notations. If Δ (=P) is reflexive, it is very natural (and, for some applications, more natural) to consider its dual $\Delta^*\subset N_\mathbb{R}$ as well. If PALP should interpret the input as Δ ^* , it must be instructed to do that by an option (`-D' for poly.x and mori.x, `-N' for nef.x). In fact, in the case of mori.x it would be extremely unnatural to use Δ as input; therefore matrix input is allowed only with `-D' to avoid errors.

A second input format uses the fact that many polytopes (in particular the ones related to the toric description of weighted projective spaces) afford a description as the convex hull of all points X that lie in the (n − 1)-dimensional sublattice $M\subset\mathbb{Z}^n$ determined by $\sum_{i=1}^n w_iX_i=0$ and satisfy the inequalities $X_i\ge -1$ for $i\in\{1,\ldots,n\}$. Given such a weight system in the format d w1 w2 ... wn where the w_i must be positive integers and $d=\sum_{i=1}^n w_i$, PALP computes the corresponding list of points and makes a transformation to $M\simeq \mathbb{Z}^{n-1}$. The following example corresponds to the Newton polytope of the quintic threefold in $\mathbb{P}^4$.

palp$ poly.x -v
Degrees and weights  `d1 w11 w12 ... d2 w21 w22 ...'
  or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
5 1 1 1 1 1
4 5  Vertices of P
   -1    4   -1   -1   -1
   -1   -1    4   -1   -1
   -1   -1   -1    4   -1
   -1   -1   -1   -1    4

As the first line of the prompt indicates, this format can be generalized to the case of k weight systems describing a polytope in $M\simeq \mathbb{Z}^{n-k}$. We call the corresponding data, which should satisfy $w_{ij}\ge 0$ and $(w_{1j},\ldots, w_{kj})\ne (0,\ldots,0)$, a CWS ('combined weight system').

It is also possible to specify a sublattice of finite index corresponding to the condition $\sum_{i=1}^nl_ix_i=0$ mod r by writing /Zr: l1 ...ln after the specification of the (C)WS.

In the following example, 21100 20011 describes a square whose edges have lattice length 2, whereas the condition indicated by Z2: 1 0 1 0 eliminates the interior points of the edges. The particular output arises because PALP transforms the original and the reduced lattice to $\mathbb{Z}^2$ in different ways.

palp$ poly.x -v
Degrees and weights  `d1 w11 w12 ... d2 w21 w22 ...'
  or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
2 1 1 0 0  2 0 0 1 1
2 4  Vertices of P
   -1    1   -1    1
   -1   -1    1    1
Degrees and weights  `d1 w11 w12 ... d2 w21 w22 ...'
  or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
2 1 1 0 0  2 0 0 1 1  /Z2: 1 0 1 0
2 4  Vertices of P
   -1    0    0    1
    1   -1    1   -1

If PALP is used interactively, it can be terminated by entering an empty line instead of the data of a polytope. In the case of file input the end of the file results in the termination.