Paley type group schemes and planar Dembowski–Ostrom polynomials

Special Mappings of Finite Fields

“…Surprisingly, for Dembowski-Ostrom polynomials, also the inverse of Theorem 5.3 holds as independently shown in [37,42,99]: 37]). Let P : F q → F q be given by a Dembowski-Ostrom polynomial.…”

“…This is false, since obviously the latter property is fulfilled for any sum O + L of a Dembowski-Ostrom polynomial O with a linearized one L. The statement of Theorem 5.4 does not hold in general for such sums O + L as shown in [75], answering negative Questions 2.5 and 2.6 from [37].…”

“…The authors of [37] assume that the Dembowski-Ostrom polynomials O(x) = n−1 i,j=0 a i,j X p i +p j are exactly those for which the difference mappings…”

Permutation polynomials over finite fields — A survey of recent advances

“…When g(X 2 ) is a Dembowski-Ostrom polynomial, i.e., of the form a ij X p i +p j ∈ F q [X], where p = char F q , Chen and Polhill [16] proved that the converse of Theorem 6.13 also holds. Theorem 6.14.…”

“…Theorem 6.14. (See Chen and Polhill [16].) Let q ≡ −1 (mod 4), and let f ∈ F q [X] be a Dembowski-Ostrom polynomial.…”