On two problems of Carlitz and their generalizations

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.