Binary quadratic and cubic forms, and unipositive matrices

Abstract

The classical method of reducing a positive binary quadratic form to a semi-reduced form applies translations alternately left and right to minimize the absolute value of the middle coefficient — and may therefore be called absolute reduction. There is an alternative method which keeps the sign of the middle coefficient constant before the end: we call this method positive reduction. Positive reduction seems to make possible an algorithm for finding the representations of 1 by a binary cubic form with real linear factors, and has various properties somewhat simpler than those of absolute reduction. Some of these properties involve unipositive matrices (with nonnegative integer elements and determinant 1). Certain semigroups of unipositive matrices with unique factorization into primes are described. Two of these semigroups give a neat approach to the reduction of indefinite binary quadratic forms—which may generalize. Some remarks on unimodular automorphs occur in Section 6.