Extremal property of some surfaces in $n$-dimensional Euclidean space

V. I. Bernik ; É. I. Kovalevskaya

Matematičeskie zametki, Tome 15 (1974) no. 2, pp. 247-254.

Résumé

A surface $\Gamma(f_1(x_1,\dots,x_m),\dots,f_n(x_1,\dots,x_n))$ is said to be extremal if for almost all points of $\Gamma$ the inequality $$\|\alpha_1f_1(x_1,\dots,x_m)+\dots+\alpha_nf_n(x_1,\dots,x_n)\|^{-n-\varepsilon},$$ where $H=\max(|\alpha_i|)$, ($i=1,2,\dots,n$), has only a finite number of solutions in the integers $\alpha_1,\dots,\alpha_n$. In this note we prove, for a specific relationship between $m$ and $n$ and a functional condition on the functions $f_1,\dots,f_n$, the extremality of a class of surfaces in $n$-dimensional Euclidean space.