An improved bound on the number of point-surface incidences in three dimensions

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Analysis shows O(m^2k^3k-1n^3k-33k-1+m+n) point-surface incidences in three dimensions, indicating tighter constraints on smooth surfaces.

The study provides a new bound, revealing O((mn)^(3/4)+m+n) incidences with spheres in three dimensions, improving upon previous estimates.
It derives this bound using the discrete polynomial ham sandwich theorem, adapting R^3 into suitable open cells.
The findings apply to point-surface incidences, showcasing advancements in our understanding of algebraic surfaces' intersection behavior.
The work includes applications to unit distances among m points in R^3, highlighting practical implications of the results.

Joshua Zahl (2013) studied this question.

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Analysis shows O(m^2k^3k-1n^3k-33k-1+m+n) point-surface incidences in three dimensions, indicating tighter constraints on smooth surfaces.