On binary quadratic forms with semigroup property

Abstract

A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x,y))=f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f has semigroup property. We give an explicit description of all pairs (f,s) with the property stated above. We do not know any other examples of forms with semigroup property.