A conditional bound on sphere tangencies in all dimensions

We use polynomial method techniques to bound the number of tangent pairs in a collection of \(N\) spheres in \(\mathbb{R}^n\) subject to a non-degeneracy condition, for any \(n \geq 3\). The condition, inspired by work of Zahl for \(n=3\), asserts that on any sphere of the collection one cannot have more than \(B\) points of tangency concentrated on any low-degree subvariety of the sphere. For collections that satisfy this condition, we show that the number of tangent pairs is \(O_{\epsilon}(B^{1/n - \epsilon} N^{2 - 1/n + \epsilon})\).