Abstract
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which yields similar applications. In this paper we show that certain functions over \({\mathbb{F}_{2^{r}}}\) are planar, which proves a conjecture of Schmidt and Zhou. The key to our proof is a new result about the \({\mathbb{F}_{q^{3}}}\) -rational points on the curve x q-1 + y q-1 = z q-1.
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The authors were partially supported by NSF grant DMS-1162181.
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Scherr, Z., Zieve, M.E. Some Planar Monomials in Characteristic 2. Ann. Comb. 18, 723–729 (2014). https://doi.org/10.1007/s00026-014-0248-3
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DOI: https://doi.org/10.1007/s00026-014-0248-3