## 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)

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