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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 $n=3$.

kklosteer
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    I think calling it a hyper-plane isn't necessary because a plane represents any surface 2D or above and I think higher dimensional circles are called hyper-spheres. As for the cubic polynomial you can factorise this – Henry Lee Jan 03 '21 at 01:48
  • Thanks for the suggestions, @HenryLee -- I've edited the question for clarity. As for factoring, I had noticed that the polynomial can be expressed as $$\sum_{j=1}^n x_j(x_j^2 + b_j) = B$$ but I am not able to see how that leads to a solution -- am I missing something? – kklosteer Jan 03 '21 at 04:57
  • Are there any conditions on any of the constants? Are any of them known to be non-negative, for instance? – Brian Tung Jan 03 '21 at 20:57
  • I am hoping for insight for the general case, but I am also interested specifically in the case that $b_j \geq 0$ for each $j$, @BrianTung. – kklosteer Jan 04 '21 at 23:41

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