Small solutions of quadratic congruences, II

Small Solutions of Quadratic Congruences, and Character Sums With Binary Quadratic Forms

“…, x n ] be a quadratic form. This paper, which may be seen as a continuation of the author's earlier work [10], [11] seeks to understand the smallest solution of the congruence Q(x) ≡ 0 (mod q) in non-zero integers x. Thus we shall set m(Q; q) := min{||x|| : x ∈ Z n − {0}, Q(x) ≡ 0 (mod q)} where ||x|| denotes the Euclidean norm, and ask (in the first instance) about…”

“…where the maximum is taken over all integral quadratic forms in n variables. (This definition differs slightly from that used in [10] and [11].) The interested reader may refer to Baker [1,Chapter 9] for an account of this problem and its applications.…”

“…(see [11,Theorem 2]); and that B n (q) ≪ ε,n q 1/2+3/n 2 +ε for every even n ≥ 2 (see [11,Theorem 3]). Indeed a number of other such bounds are possible.…”

“…We now have the same exponent 5/8 for (non-singular) forms in 3 variables as we previously had for 4 variables. (However it is explained in [11] that one can reduce the exponent to 13/21 with more work, in the 4 variable case. )…”

Small solutions for system of homogenous polynomials congruences over a Dedekind domain

“…Dealing with m = p, p an odd prime, Heath-Brown [14] obtained a nonzero solution of (2.1) with max |x i | p 1/2 log p for n 4. His result was an improvement on the result of [15] in this case.…”