We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products \(K = \displaystyle \prod \nolimits _{i = 1}^d K_i\), where \(K_i \subseteq \mathbb {C}\) are compact sets and \(\mathbb {C} {\setminus } K_i\) are connected, \(i = 1, \ldots , d\) and \(0 \notin K\). On such K the partial sums approximate uniformly any polynomial. Finally, the partial sums may be replaced by more general expressions. The phenomenon is topologically and algebraically generic.

我们将谢列兹涅夫的通用幂级数归纳为几个变量，并允许系数取决于参数。然后，近似函数可以取决于相同的参数。乘积\（K = \ displaystyle \ prod \ nolimits _ {i = 1} ^ d K_i \）成立，其中\（K_i \ subseteq \ mathbb {C} \）是紧集，而\（\ mathbb { C} {\ setminus} K_i \）已连接，\（i = 1，\ ldots，d \）和\（0 \ notin K \）。在这样的K上，部分和统一一致地近似任何多项式。最后，部分和可以用更通用的表达式代替。这种现象在拓扑和代数上都是通用的。