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We prove a Szemer\'edi-Trotter type theorem and a sum-product estimate in thesetting of finite quasifields. These estimates generalize results of the fourthauthor, of Garaev, and of Vu. We generalize results of Gyarmati and S\'ark\"ozyon the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over afinite field. Other analogous results that are known to hold in finite fieldsare generalized to finite quasifields.

A Szemer\'edi-Trotter type theorem, sum-product estimates in finite quasifields, and related results