If k of them $ x_1,...,x_k$ negative, then $ x_{k+1}+...+x_{1997}>1997$. Let \[ T_n=\frac{\sum_{i=k+1}^{1997}x_i^n}{1997-k}, S_n=\sum_{i=1}^{1997}x_i^n.\] Then $ S_4\ge (1997-k)T_4\ge (1997-k)(T_3)^{4/3}\ge (1997-k)(T_1)^4$. Equality only if $ k=0$ and $ x_i=1$ for all i.