On some equations over finite fields

45 - On some equations over finite fields par IOULIA BAOULINA

On some permutation polynomials over finite fields

Let Fq be a finite field of q = pm elements with characteristic p. A polynomial P(x) ∈ Fq[x] is called a permutation polynomial of Fq if P(x) induces a bijective map from Fq to itself. In general, finding classes of permutation polynomials of Fq is a difficult problem (see [3, Chapter 7] for a survey of some known classes). An important class of permutation polynomials consists of permutation p...

On the Carlitz problem on the number of solutions to some special equations over finite fields

We consider an equation of the type a1x 2 1 + · · ·+ anxn = bx1 · · ·xn over the finite field Fq = Fps . Carlitz obtained formulas for the number of solutions to this equation when n = 3 and when n = 4 and q ≡ 3 (mod 4). In our earlier papers, we found formulas for the number of solutions when d = gcd(n − 2, (q − 1)/2) = 1 or 2 or 4; and when d > 1 and −1 is a power of p modulo 2d. In this pape...

Discrete Integrable Equations over Finite Fields

Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang–Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the ...

Certain diagonal equations over finite fields

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