Let $$G$$ be the companion matrix of the $$m$$th cyclotomic polynomial, of degree $$n = \phi(m)$$, generating the standard $$n$$-dimensional representation of the cyclic group $$C_m$$. A cyclotomic form is a real quadratic form $$q(x)$$ in $$n$$ variables, invariant under this action of $$C_m$$: if $$A$$ is the symmetric matrix of $$q$$, then $$G^tAG = G$$. Voronoi’s algorithm is a long-known method providing an explicit complete classification of perfect quadratic forms, and thus a means of computing the densest quadratic forms in each dimension, albeit with tremendous computational complexity.
This article generalizes Voronoi’s algorithm to cyclotomic forms (which substantially reduces the computational complexity), and provides a complete classification of perfect cyclotomic forms for $$\phi(m) < 16$$ and for $$m = 17$$. In the process, precise upper bounds for the Hermite invariant of cyclotomic forms in this range are obtained. In many cases, these bounds improve the best-known or conjectured values of the Hermite constant for the corresponding dimensions.