Counting roots of fully triangular polynomials over finite fields.

Let F q be a finite field with q elements, f ∈ F q [ x 1 , ... , x n ] a polynomial in n variables and let us denote by N (f) the number of roots of f in F q n. In this paper we consider the family of fully triangular polynomials, i.e., polynomials of the form f (x 1 , ... , x n) = a 1 x 1 d 1 , 1 + a 2 x 1 d 1 , 2 x 2 d 2 , 2 + ... + a n x 1 d 1 , n ⋯ x n d n , n − b , where d i , j > 0 for all 1 ≤ i ≤ j ≤ n. For these polynomials, we obtain explicit formulas for N (f) when the augmented degree matrix of f is row-equivalent to the augmented degree matrix of a linear polynomial or a quadratic diagonal polynomial. [ABSTRACT FROM AUTHOR]