Generic polynomials of fixed degree d at least 2 on product spaces are non-Cartesian, with algorithmic decidability via Groebner bases and applications yielding sharp incidence bounds.
Zahl, An improved bound on the number of point-surface incidences in three dimensions,Contri- butions to Discrete Mathematics, 8(1):100–121
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On the structure and generic non-Cartesianity of polynomials in product spaces
Generic polynomials of fixed degree d at least 2 on product spaces are non-Cartesian, with algorithmic decidability via Groebner bases and applications yielding sharp incidence bounds.