Second minimum of inhomogeneous indefinite quadratic forms of signature \(\pm 2\) (Q5936403)

The paper considers \(n\)-ary real indefinite quadratic forms of type \((r,n-r)\) and determinant \(D \not= 0\). Let \(C_{r,n-r}\) denote the infimum of all positive numbers \(C\) such that for any quadratic form \(Q\) of the mentioned type and determinant and real numbers \(c_1,\dots,c_n\) there exist \((x_1,\dots,x_n)\equiv(c_1,\cdots,c_n)\pmod 1\) satisfying \[ |Q(x_1, \dots,x_n)|\leq(C|D|)^{\frac{1}{n}}. \] At present the minima \(C_{r,n-r}\) and \(k\)th successive inhomogeneous minima \(C_{r,n-r}^{(k)}\) have been determined in various cases and by various mathematicians. In the present paper the authors continue their previous investigations and prove that \(C_{r+2,2}^{(2)}=C_{r,r+2}^{(2)}=\frac{1}{4}\) for all \(r\geqslant 1\). The paper determines also all the critical forms.