A note on Siegel's lemma over number fields
Let \(K\) be a number field of degree \(d\) and discriminant \(D_K\). Define \(\Delta_K = \sqrt {|D_K |}\). The height \(H(a)\) of a vector \({\mathbf a} \in K^n\) is the projectively invariant absolute Weil height. Define \(H'(a)\) in the same way, but with the Euclidean norm at archimedean places instead of the sup norm. The inhomogeneous height \(H_1 ({\mathbf a})\) is \(H(1, {\mathbf a})\), while the height \(H'(A)\) of a matrix \(A\) with rows \({\mathbf a}_1, \dots, {\mathbf a}_M\) is \(H'( {\mathbf a}_1 \wedge \cdots \wedge {\mathbf a}_M)\). Let \(S\) be a system of \(M\) linearly independent homogeneous linear forms in \(N\) variables with coefficients in \(K\). The height \(H'(S)\) of the system \(S\) is the height \(H'(A)\) of its coefficient matrix \(A\). Assume \(N > M\). \textit{E. Bombieri} and \textit{J. Vaaler} have shown [Invent. Math. 73, 11-32 (1983; Zbl 0533.10030)] that there exists a basis \({\mathbf x}_1, \dots, {\mathbf x}_{N - M} \) of the solution space of \(S\) in \(K^N\) such that \[ \prod^{N - M}_{i = 1} H ({\mathbf x}_i) \leq \bigl( (2/ \pi)^{rsb2} \Delta_K \bigr)^{(N - M)/d} H'(S). \] A similar estimate holds (with the inhomogeneous height \(H_1)\) for a basis of the solution space of \(S\) in the ring of integers \({\mathcal O}_K\) of \(K\). The authors show that these estimates are sharp. For \(d,N,M\) positive integers with \(d > 1\) and \(N \geq M + 2\), there exists a constant \(C_1 > 0\) depending only on \(d\) and \(N - M\) such that, for any number field \(K\) of degree \(d\), there exists a system \(S\) of \(M\) linearly independent homogeneous linear forms in \(N\) variables with coefficients in \(K\) for which any basis \({\mathbf x}_1, \dots, {\mathbf x}_{N - M}\) of the solution space in \(K^N\) satisfies \[ \prod^{N - M}_{j = 1} H ({\mathbf x}_i) \geq X\quad\text{where}\quad X = C_1 \Delta_K^{\bigl[ (N - M - 1)/2 \bigr] /d(d - 1)} H'(S). \] As pointed out by D. W. Masser to the authors (private communication), their result implies that any nonzero solution \({\mathbf x}\) in \(K^N\) of \(S\) satisfies \(H({\mathbf x}) \geq X^{1/(N - M)}\). Again, similar statements hold for solutions in the ring of integers of \(K\) and the inhomogeneous height \(H_1\). It follows from the previous result that a power of \(\Delta_K\) is needed in the upper bound of Bombieri and Vaaler, at least when \(N \geq M + 3\). The authors also show that the same is true when \(N = M + 2\), and they refer to \textit{D. W. Masser} for the remaining case \(N = M + 1\) [A note on Siegel's lemma, Rocky Mt. J. Math. (to appear)]. The paper under review concludes with a surprising example showing that the dependence on the discriminant may disappear when solutions are considered in a larger field. An improvement of the above mentioned results has been obtained by \textit{J. L. Thunder} [An adelic Minkowski-Hlawka theorem and an application to Siegel's lemma, J. Reine Angew. Math. (to appear)]. The Corollary of theorem 5 of that paper shows that, for any number field \(K\) of degree \(d\) and \(N > M + 1\), there is a system of \(M\) homogeneous linear equations in \(N\) variables with coefficients in \(K\) where any basis of solutions \({\mathbf x}_1, \dots, {\mathbf x}_{N - M} \in K^N\) (in fact any \(N - M\) nonzero solutions) satisfies \[ \prod^{N - M}_{i = 1} H ({\mathbf x}_i) \geq C(N - M, d) \Delta_K^{(N - M)/d} \bigl( h(K) R(K) \bigr)^{- 1/d} H'(S). \] Here, \(C(N - M,d)\) is a constant depending only on \(N - M\) and \(d\), while \(h(K)\) and \(R(K)\) are the class number and regulator, respectively, of \(K\). The exponent on the discriminant matches that in the Bombieri-Vaaler theorem, though the class number and regulator give roughly a \(\Delta_K^{-1/d}\) term by Brauer-Siegel. (Also submitted to M. R.).
scientific article; zbMATH DE number 838725