Quadratic forms which have only large zeros
Let Q(\({\mathfrak x})=\sum_{i,j}Q_{ij}x_ ix_ j\) be a symmetric quadratic form in n variables with rational integer coefficients. Cassels proved that if Q has a non-trivial integral zero, then it has one satisfying \(| {\mathfrak x}| \ll | Q|^{(n-1)/2}.\) Here \(| {\mathfrak x}|\) and \(| Q|\) are the maximum modulus of the coordinates (respectively coefficients) of \({\mathfrak x}\) and Q, and the \(\ll\) constants will always depend only on n. Kneser showed that Cassel's result is best possible. In earlier papers the authors showed that if Q is non-singular and vanishes on a d-dimensional rational subspace for some \(n\geq 2d>0\), then Q has a non-trivial integral zero satisfying \(| {\mathfrak x}| \ll | Q|^{(n-d)/2d}.\) The first theorem in this paper constructs quadratic forms which show that this higher dimensional analogue of Cassel's theorem is best possible. Next, the authors consider the question of finding n linearly independent integral zeros of Q. Earlier they proved that if Q is non-singular and vanishes on a d-dimensional rational subspace with \(n\geq 2d>0\), then there are n linearly independent integral zeros \({\mathfrak x}_ 1,...,{\mathfrak x}_ n\) of Q satisfying \[ (*)\quad | {\mathfrak x}_ 1| \cdot.... \cdot | {\mathfrak x}_ n| \quad \ll \quad | Q|^{(n^ 2/2d)-d}. \] The second theorem in this paper gives a construction which shows that (*) is best possible. In particular, Davenport had conjectured that (*) should hold with exponent \((n^ 2/2d)-(n/2)\); the authors' estimate shows that Davenport's conjecture is only true in the case \(n=2d\).
higher dimensional analogue of Cassel's theorem
scientific article; zbMATH DE number 4120246