I am trying to determine conditions under which three surfaces in $\mathbb{R}^n$ intersect.
The first two surfaces are an $n$-plane and an $n$-sphere: $$\sum_{j=1}^n x_j = C$$ $$\sum_{j=1}^n x_j^2 = R^2$$ These intersect in an $(n-1)$-sphere so long as the radius $R$ is greater than the distance from the origin to the hyper-plane. Because the plane's normal vector is $(1, \cdots, 1)$, this distance is simply $\tfrac{C}{\sqrt{n}}$.
I am trying to determine conditions which would guarantee that this $(n-1)$-sphere intersects with a third surface: $$\sum_{j=1} (x_j^3 + b_j\cdot x_j) = B$$ for constants $b_j$ and $B$.
I would be grateful for any insights or suggestions, even for the special case $n=3$.