Systems of diagonal Diophantine inequalities

We treat systems of real diagonal forms $F_1({\mathbf x}), F_2({\mathbf x}), \ldots, F_R({\mathbf x})$ of degree $k$, in $s$ variables. We give a lower bound $s_0(R,k)$, which depends only on $R$ and $k$, such that if $s \geq s_0(R,k)$ holds, then, under certain conditions on the forms, and for any positive real number $\epsilon$, there is a nonzero integral simultaneous solution ${{\mathbf x}\in {\mathbb Z}^s}$ of the system of Diophantine inequalities $\vert F_i({\mathbf x})\vert < \epsilon$ for $1 \leq i \leq R$. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.