Representation of numbers by quadratic forms in algebraic number fields

Abstract

Let K be the totally real field of algebraic numbers of degree n=[k∶2e] with the discriminant D=D(K); t=t(x1, ..., xs) a totally positive quadratic form of the determinant d>0 over the ring σ of integers from the field K; S⩾4. Let be the number of representations over σ of the number m∈σ by the form a complete singular series. It is proved that for given s and n, there exists a constant c such that for N(d)>0 it is not true that for all m∈σ with m totally positive.