Counting roots of fully triangular polynomials over finite fields

Abstract

Let Fq be a finite field with q elements, f∈Fq[x1,…,xn] a polynomial in n variables and let us denote by N(f) the number of roots of f in Fqn. In this paper we consider the family of fully triangular polynomials, i.e., polynomials of the formf(x1,…,xn)=a1x1d1,1+a2x1d1,2x2d2,2+…+anx1d1,n⋯xndn,n−b, where di,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.