Counting roots of fully triangular polynomials over finite fields

Let Fq be a finite field with q elements, f is an element of Fq[x1, ... , xn] 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(x1,...,xn)=a1xd1,11+a2xd1,21xd2,22+<middle dot><middle dot><middle dot>+anxd1,n1<middle dot><middle dot><middle dot>xdn,n 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.(c) 2023 Elsevier Inc. All rights reserved.