On two problems of Carlitz and their generalizations

TL;DR

This paper investigates the number of solutions to a specific polynomial equation over finite fields, extending Carlitz's formulas and providing explicit solutions in new cases, thereby advancing understanding of polynomial equations in finite field theory.

Contribution

The paper extends Carlitz's results by explicitly determining the solution count for more general cases of the polynomial equation over finite fields.

Findings

Derived explicit formulas for solution counts in new cases

Extended previous results to broader parameter settings

Confirmed the solution count formula under specific gcd conditions

Abstract

Let $N_q$ be the number of solutions to the equation $$ (a_1^{}x_1^{m_1}+\dots+a_n^{}x_n^{m_n})^k=bx_1^{k_1}\cdots x_n^{k_n} $$ over the finite field $\mathbb F_q=\mathbb F_{p^s}$. Carlitz found formulas for~$N_q$ when $k_1=\dots=k_n=m_1=\dots=m_n=1$, $k=2$, $n=3$ or $4$, $p>2$; and when ${m_1=\dots=m_n=2}$, $k=k_1=\dots=k_n=1$, $n=3$ or $4$, $p>2$. In earlier papers, we studied the above equation with $k_1=\dots=k_n=1$ and obtained some generalizations of Carlitz's results. Recently, Pan, Zhao and Cao considered the case of arbitrary positive integers $k_1,\dots,k_n$ and proved the formula $N_q=q^{n-1}+(-1)^{n-1}$, provided that $\gcd(\sum_{j=1}^n (k_jm_1\cdots m_n/m_j)-km_1\cdots m_n,q-1)=1$. In this chapter, we determine $N_q$ explicitly in some other cases.