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On some equations over finite fields

Ioulia Baoulina — 2005

Journal de Théorie des Nombres de Bordeaux

In this paper, following L. Carlitz we consider some special equations of n variables over the finite field of q elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on n and q .

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

Ioulia N. Baoulina — 2011

Journal de Théorie des Nombres de Bordeaux

We consider an equation of the type a 1 x 1 2 + + a n x n 2 = b x 1 x n over the finite field 𝔽 q = 𝔽 p s . 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  2 d . In this paper, we obtain formulas for the number of solutions when d = 2 t , t 3 , p 3 or 5 ( mod 8 ) or p 9 ( mod 16 ) . For general case, we derive lower bounds for the number of solutions.

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