Small solutions of quadratic Diophantine equations (Q2756983)

The author considers the solubility of the Diophantine equation NEWLINE\[NEWLINE\sum_{1 \leq i,j \leq s} a_{ij} x_i x_j + \sum_{i=1}^s h_i x_i =n \tag \(*\) NEWLINE\]NEWLINE with given integer coefficients \(a_{ij}, h_i,n\). It is no serious restriction to assume that \(a_{ij}=a_{ji}\) and \(\det (a_{ij}) \not=0\). Let \(H=\max \{|a_{ij}|, |h_i|, |n |\}\). \textit{C. L. Siegel} proved [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1972, 21--46 (1972; Zbl 0252.10019)] that there is an effectively computable function \(\Lambda_s(H)\) with the property that if the quadratic form \((*)\) admits a solution in integers, then there is a solution with \(\max |x_i |\leq \Lambda_s(H)\). It was proved by \textit{D. Kornhauser} [Acta Arith. 55, 83--94 (1990; Zbl 0705.11010); Math. Proc. Camb. Philos. Soc. 107, 197--211 (1990; Zbl 0709.11022)] that for \(s=2\) there is an exponential bound \(\Lambda_2(H) = (c_1 H)^{c_2 H}\) and for \(s\geq 5\) there is a polynomial bound \(\Lambda_s(H)=c_3(s)H^{c_4(s)}\). Here the \(c_i\) are positive constants. These bounds are not too far from being best possible. NEWLINENEWLINENEWLINEIn the thesis under review, the author proves a polynomial bound for the remaining cases with \(s=3\) or \(s=4\) variables. Moreover he gives improvements for \(s \geq 5\). The new bounds are, with effectively computable constants \(C_i\): NEWLINE\[NEWLINE\Lambda_s(H)= \begin{cases} C_3 H^{2100}& \text{ when } s=3\\ C_4 H^{84}& \text{ when } s=4\\ C_sH^{5s+19+74/(s-4)}& \text{ when } s \geq 5. \end{cases} NEWLINE\]NEWLINE For \(s = 4\) the proof rests on the circle method and ideas from the geometry of numbers. For the most difficult case \(s=3\), the author shows in an ingenious manner that this problem can (surprisingly) be reduced to the case with \(s=4\) variables.