A normal form for definite quadratic forms over \(\mathbb{F}_{q}[t]\) (Q2894524)

Let \(q\) be a power of an odd prime, \(\mathbb F_q\) the finite field with \(q\) elements, \(\mathbb F_q[t]\) the ring of polynomials in one variable and \(\mathbb F_q(t) = \mathrm{frac}(\mathbb F_q[t])\) the field of rational functions over~\(\mathbb F_q\). Let \(Q\) be a definite quadratic form on a finite dimensional \(\mathbb F_q(t)\)-space \(V\).NEWLINENEWLINE\textit{D. Ž. Đoković} [J. Algebra 43, 359--374 (1976; Zbl 0343.15006)] introduced the notion of a reduced basis of an \(\mathbb F_q[t]\)-lattice in a quadratic space \((V,Q)\). \textit{L. J. Gerstein} [J. Algebra 268, No. 1, 252--263 (2003; Zbl 1042.11025)] showed that such reduced bases achieve successive minima and their number is finite. However, reduced Gram matrices are not in a bijective correspondence with the isometry classes of lattices.NEWLINENEWLINEThe goal of the present paper is to remedy this situation by introducing a so-called \textit{normal Gram matrix}, which distinguishes isometry classes. In other words, the author shows that two lattices are isometric if and only if they have the same normal Gram matrices. He then demonstrates an efficient algorithm for computing such a normal Gram matrix starting with any reduced Gram matrix. His algorithm has already been implemented in \texttt{MAGMA} and several other computer algebra systems.