Hi.

For a matrix $D \in \mathbb{Z}^{n \times n}$ and a symmetric, positive definite integral even matrix $S \in \mathbb{Z}^{n \times n}$ put $S[D] := D^TSD$ where the $\cdot^T$ means 'transposed'. Further, for a symmetric matrix $X \in \mathbb{Z}^{n \times n}$ put $$R(S, X) := \{ D \in \mathbb{Z}^{n \times n} : S[D] = X \}$$

My question is the following: Can one give a more concrete description of the elements in it or can one at least compute this set in an efficient way (for relatively small $n$, say $\leq 20$ or so)? If this is not possible/unknown, does it maybe work with

$$R(S, X) / GL_n(\mathbb{Z})$$

or something like this? From computations in very small dimensions i got the impression that this set is actually relatively small.

One can show (see for example Koecher/Krieg, a german book on modular forms, p. 262) that $R(S, X)$ is finite and that

$$R(S, X) \subset \{ D \in \mathbb{Z}^{n \times n} : \sum_{i,j} |D_{i,j}|^2 \leq \alpha(S,X) + 1 \}$$

where $\alpha(S,X) = Trace(X)/\lambda$ and $\lambda$ is the minimal eigenvalue of $S$, i.e. $\alpha(S,X)$ can be easily computed. The problem is, of course, that the right hand set is pretty big even for dimensions $\leq 20$.

Cheers,

Fabian Werner