Planarity of products of two linearized polynomials

Abstract Let L 1 ( x ) and L 2 ( x ) be linearized polynomials over F q n . We give conditions when the product L 1 ( x ) ⋅ L 2 ( x ) defines a planar mapping on F q n . For a polynomial L over F q n , let M ( L ) = { α ∈ F q n : L ( x ) + α ⋅ x is bijective on F q n } . We show that the planarity of the product L 1 ( x ) ⋅ L 2 ( x ) is linked with the set M ( L ) of a suitable linearized polynomial L . We use this relation to describe families of such planar mappings as well as to obtain nonexistence results.

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