HELP FOR: uforms/unit_form

SYNOPSIS:

• A unit form q is a map
  Zⁿ ---> Z,

  x --->

  xi2 +

  qi,jxixj

  where Z is the ring of integers and q_i,j are integers.

• A unit form q is represented by its symmetric (n x n) matrix Q with coefficients Q_i,i=2 for i=1..n, Q_i,j=Q_j,i =q_i,j for 1<=i<j<=n.
  It follows that q(x) = 1/2 x Q x^T.

• We can also treat a unit form as a polynomial in n variables.

• There is a well-known bijection between the set of unit forms in n variables and the set of bigraphs on the vertex set {1,..,n} (possibly with multiple edges, but without loops) defined as follows: For i<>j the vertex i is connected with vertex j with -q_i,j solid edges, if q_i,j<0, and with q_i,j dotted edges, if q_i,j>0.
  And of course for each bigraph (without loops) with numerated vertices we have a corresponding unit form in an inverse way, setting a_i,i:=2, i=1..n, a_j,i:=a_i,j, 1<=i<j<=n.

• For simplicity of description we will in several cases treat a unit form as a corresponding bigraph.