## Bacher Polynomials

The Bacher polynomials associated with a lattice L are defined as follows.

With each vector v in L of minimal norm m we associate a univariate polynomial B_v in Z[X] in the following way. Let M_v be the set of minimal vectors in L that have scalar product m/2 with v. For each w in M_v let n_w be the number of pairs (x,y) in M_v x M_v such that all scalar products b(w,x)=b(w,y)=b(x,y)=m/2. Then B_v := sum _{w in M_v} X^n_w.

The Bacher polynomials are a very strong invariant of the lattice and provide a powerful method for distinguishing between lattices. They often separate the orbits of the automorphism group on the set of minimal vectors of L.

In the files we give a representative v for each orbit of the automorphism group of the lattice on the minimal vectors followed by the polynomial. The number in brackets indicates the number of pairs {v,-v} lying in this orbit. So the sum of these numbers is half of the KISSING_NUMBER of the lattice.

These polynomials were first used in the thesis of Roland Bacher,
R'eseaux unimodulaires sans automorphismes, Th`ese, Universit'e de Gen`eve 1993,
and are also described in
W. Plesken, B. Souvignier, Computing isometries of lattices. J. Symb. Computation (to appear)