Small Solutions of Congruences Original Research Article

Let Qj(λ) = Qj(λ1, ..., λs) (1 ≤ j ≤ h) be a system of quadratic forms with coefficients in integers of an algebraic number field K of degree n, and let image be an integral ideal of K. The purpose of the paper is to prove that the system of congruences Qj(λ) ≡ 0 (mod image) (1 ≤ j ≤ h) has a nonzero solution λ, satisfying maxi, jλ(i)j much less-than N(image)1/2 + ε provided that s ≥ c(h, n, ε). This improves a result of T. Cochrane (1987, Illinois J. Math.31, 618-625) and also gives a generalisation of a result of R. C. Baker (1980, Mathematika27, 30-45). The small solutions of additive congruences are also considered.