Extending Djoković's lattice reduction algorithm to include isotropic lattices (Q2438013)

Let \(k\) be a field of characteristic not \(2\). According to a theorem of \textit{D. Ž. Đoković} [J. Algebra 43, 359--374 (1976; Zbl 0343.15006)], every \(k[x]\)-lattice \(L\) on an anisotropic quadratic space \(V\) over \(k(x)\) has a basis that is reduced, in the sense that the associated Gram matrix \((a_{ij})\) satisfies the two conditions i) \(\partial a_{11} \leq\ldots \leq \partial a_{nn}\), and ii) \(\partial a_{ii} > \partial a_{ij}\) for \(j \neq i\), where \(\partial\) denotes the degree function on \(k(x)\). The main theorem of the paper under review states that this result remains true when the assumption that \(V\) is anisotropic is removed. That is, every regular \(k[x]\)-lattice has a reduced basis. The author points out several new results and alternate proofs of known results that are consequences of this extension of Đoković's theorem. A lattice reduction algorithm based on Đoković's proof was used by \textit{L. J. Gerstein} [J. Algebra 268, No. 1, 252--263 (2003; Zbl 1042.11025)] to prove that the classification of definite \(\mathbb F_q[x]\)-lattices reduces to a problem over the finite field \(\mathbb F_q\), when \(q\) is odd. That is, if \(L\) and \(M\) are definite quadratic \(\mathbb F_q[x]\)-lattices of rank \(n\) having associated Gram matrices \(A\) and \(C \in M_n(\mathbb F_q[x])\), respectively, then \(L\) and \(M\) are isometric if and only if there exists \(T\in\mathrm{GL}_n(\mathbb F_q)\) such that \(C={}^tTAT\). It is shown by an example in the present paper that the corresponding result does not hold for indefinite lattices.