0 Introduction and the main result

Markoff-Hurwitz-type equations are the following type of the Diophantine equation:

$ x_1^2 + x_2^2 + \cdots + x_n^2 = c{x_1}{x_2} \cdots {x_n}, $

where n, c are positive integers and n≥3. This type of equations were firstly studied by MARKOFF^[1] for the case n=3, c=3 because of its relation to Diophantine approximation. More generally, these equations were studied by HURWITZ ^[2].

CARLITZ^[3] investigated the Markoff-Hurwitz-type equations over finite fields. Let N_q denote the finite of q elements and $\mathbb{F}$_q^*=$\mathbb{F}$_q{0}, He investigated the number of $\mathbb{F}$_q-rational points on the equation

$ {a_1}x_1^2 + {a_2}x_2^2 + \cdots + {a_n}x_n^2 = c{x_1}{x_2} \cdots {x_n}, $

where a₁, a₂, …, a_n, c∈$\mathbb{F}$_q^* and n≥3, and obtained the formulas in terms of Jacobsthal sum for n=3 and n=4. CARLITZ^[4] studied equations of the form (x₁+x₂+…+x_n)²=bx₁x₂…x_n over finite fields of odd characteristic.

Recently, BAOULINA^[5-7] studied the generalized Markoff-Hurwitz-type equation:

$ {a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = c{x_1}{x_2} \cdots {x_n}, $

where a_i, c∈$\mathbb{F}$_q^*, m_i are positive integers which satisfied m_i|(q-1) for i=1, 2, …, n and n≥2. PAN et al^[8] considered the further generalized Markoff-Hurwitz-type equations of the form:

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = cx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}, $

(1)

where n≥2, m_i, k_i, k are positive integers, a_i, c∈$\mathbb{F}$_q^*, for 1≤i≤n. The special case (1) of k=1 is investigated by CAO^[9]. For positive integers d₁, d₂, …, d_k, let I(d₁, d₂, …, d_k) denote the number of k-tuples (j₁, j₂, …, j_k) of integers with 1≤j_r≤d_r-1(1≤r≤k) such that $\left( \frac{{{j}_{1}}}{{{d}_{1}}} \right)+\left( \frac{{{j}_{2}}}{{{d}_{2}}} \right)+\cdots +\left( \frac{{{j}_{k}}}{{{d}_{k}}} \right)$ is an integer. Using the function I(m_j1, m_j2, …, m_jr)(1≤j₁ < … < j_r≤n), SONG et al^[10] presented the formulas for the number of solutions of the following equation defined over $\mathbb{F}$_q:

$ x_1^{{m_1}} + x_2^{{m_2}} + \cdots + x_n^{{m_n}} = c{x_1}{x_2} \cdots {x_t} $

under some certain restrictions, where m_j|(q-1), n≥2, c∈$\mathbb{F}$_q^*, t>n. Some other generalizations of Markoff-Hurwitz-type equations were reported in [11-12].

In this paper, we consider the rational points of the further generalized Markoff-Hurwitz-type equations of the form

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = cx_1^{{k_1}}x_2^{{k_2}} \cdots x_t^{{k_t}} $

(2)

over the finite field $\mathbb{F}$_q in some other cases, where n≥2, m_i, k_j, k, and t≥n, are positive integers, a_i, c∈$\mathbb{F}$_q^*, for 1≤i≤n, 1≤j≤t.

Let f(x₁, x₂, …, x_n) and g(x₁, x₂, …, x_n) be polynomials with coefficients in $\mathbb{F}$_q. Throughout this paper, we use N_q[f(x₁, x₂, …, x_n)=g(x₁, x₂, …, x_n)] to denote the number of the rational points of the equation f(x₁, x₂, …, x_n)=g(x₁, x₂, …, x_n) in $\mathbb{F}$_q^{max(n, s)}. We also let d_i=gcd(m_i, q-1), M=lcm[m₁, m₂, …, m_n] and N_q denote the number of solutions in $\mathbb{F}$_q^t to (2) as above. Using the results of ref. [8], we will show that the expression of N_q becomes very simple under some restrictions on the exponents. Our main result is

Theorem 1   With the notation as above, if $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{k}_{i}}M}{{{m}_{i}}}-kM, q-1} \right)=1$ and d₁, d₂, …, d_n are pairwise coprime, then we have

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Clearly, N_q is independent of the coefficients a_i, c and the exponents k_n+1, …, k_t. Letting t=n, then theorem 1 reduces to the theorem of PAN et al^[8]. Theorem 1 also generalizes the main results of [10] in some other cases.

This paper is organized as follows. In section 1, we recall some useful known lemmas. In section 2, we make use of the results presented in section 1 to show theorem 1. Some interesting applications of theorem 1 will be provided as corollaries at the end.

1 Preliminary lemmas

In this section, we present some useful lemmas which are needed in section 2. Let m be a positive integer and h(x₁, x₂, …, x_r) be a polynomial with integer coefficients. We use N[h≡0(mod m)] to denote the number of the solutions of the congruence h(x₁, x₂, …, x_r)≡(mod m). We first recall two well known results in the elementary number theory.

Lemma 1^[13]   Let a, b be positive integers. Then

$ \gcd \left( {a,b} \right){\rm{lcm}}\left[ {a,b} \right] = ab. $

Lemma 2^[13]   Let a₁, a₂, …, a_r, m, b be positive integers. Then the necessary and sufficient condition for the congruence a₁x₁+a₂x₂+…+a_rx_r≡b(mod m) to have a solution is that d=gcd(a₁, a₂, …, a_r, m)|b. If this condition is satisfied, then the number of incongruent (mod m) solutions is m^r-1d. Furthermore, we have

$ N\left[ {\sum\limits_{i = 1}^r {{a_i}{x_i}} \equiv b\left( {\bmod m} \right)} \right] = N\left[ {d\sum\limits_{i = 1}^r {{x_i} \equiv b\left( {\bmod m} \right)} } \right]. $

Lemma 3   Let t₁, t₂, …, t_r be positive integers and d=gcd(t₁, t₂, …, t_r, q-1). Then for any elements a, α∈$\mathbb{F}$_q^*, we have

$ \begin{gathered} {N_q}\left[ {ax_1^{{t_1}}x_2^{{t_2}} \cdots x_r^{{t_r}} = \alpha } \right] = {N_q}\left[ {a{{\left( {{x_1}{x_2} \cdots {x_r}} \right)}^d} = \alpha } \right] = \hfill \\ \left\{ \begin{gathered} d{\left( {q - 1} \right)^{r - 1}},\;\;{\text{if}}\;{a^{ - 1}}\alpha \;{\text{is}}\;{\text{a}}\;d - {\text{th}}\;{\text{power}}\;{\text{in}}\;\mathbb{F}_q^ * , \hfill \\ 0,\;\;{\text{otherwise}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $

Proof   Using the fact that $\mathbb{F}$_q^* is cyclic, we can let ξ be a primitive element of $\mathbb{F}$_q^*. Then for any element b∈$\mathbb{F}$_q^*, we have b=ξ^k, where k is a positive integer such that 1≤k≤q-1. Suppose that a^-1α=ξ^m, then using lemma 2, we have

$ \begin{array}{l} {N_q}\left[ {ax_1^{{t_1}} \cdots x_r^{{t_r}} = \alpha } \right] = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{\xi ^{{e_1}{t_1}}} \cdots {\xi ^{{e_r}{t_r}}} = {\xi ^m}} \right]} = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{e_1}{t_1} + \cdots + {e_r}{t_r} = m\left( {\bmod q - 1} \right)} \right]} = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {\left( {{e_1} + \cdots + {e_r}} \right)d = m\left( {\bmod q - 1} \right)} \right]} = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{{\left( {{\xi ^{{e_1}}} \cdots {\xi ^{{e_r}}}} \right)}^d} = {\xi ^m}} \right]} = \\ {N_q}\left[ {a{{\left( {{x_1}{x_2} \cdots {x_r}} \right)}^d} = \alpha } \right]. \end{array} $

Since lemma 2 tells us that

$ \begin{array}{l} \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{e_1}{t_1} + \cdots + {e_r}{t_r} = m\left( {\bmod q - 1} \right)} \right]} = \\ \;\;\;\;\;\;\left\{ \begin{array}{l} d{\left( {q - 1} \right)^{r - 1}},\;\;{\rm{if}}\;d\left| m \right.,\\ 0,\;\;{\rm{otherwise}}. \end{array} \right. \end{array} $

Then the desired result follows immediately. This ends the proof of lemma 2.

Lemma 4^[14]   Let n be a positive integer and 1≤j≤n. Assume that $\gcd \left( {{d}_{j}}, \frac{{{d}_{1}}\cdots {{d}_{n}}}{{{d}_{j}}} \right)=1$ for some j. Then

$ \begin{array}{l} {N_q}\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = 0} \right) = \\ \;\;\;\;\;\;\;\;{N_q}\left( {{a_1}x_1^{{d_1}} + {a_2}x_2^{{d_2}} + \cdots + {a_n}x_n^{{d_n}} = 0} \right) = {q^{n - 1}}. \end{array} $

The following result comes from PAN et al ^[8].

Lemma 5^[8]   In the equation (2), if t=n and $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{m}_{1}}\cdots {{m}_{n}}}{{{m}_{i}}}-k{{m}_{1}}\cdots {{m}_{n}}, q-1} \right)=1$, then we have

$ \begin{array}{l} {N_q}\left[ {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = cx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;{q^{n - 1}} + {\left( { - 1} \right)^{n - 1}}. \end{array} $

Lemma 6^{[10, 15]}   The number of elements of d-th power in $\mathbb{F}$_q^* is $\frac{\left( q-1 \right)}{d}$.

2 Proof of theorem 1

In this section, we give the proof of theorem 1.

Proof   Firstly, we claim that the condition of lemma 5 is equivalent to the conditions of theorem 1. That is, the condition

$ \gcd \left( {\sum\limits_{i = 1}^n {\frac{{{k_i}{m_1} \cdots {m_n}}}{{{m_i}}}} - k{m_1} \cdots {m_n},q - 1} \right) = 1 $

is equivalent to the following two conditions:

$ \gcd \left( {\sum\limits_{i = 1}^n {\frac{{{k_i}M}}{{{m_i}}}} - kM,q - 1} \right) = 1\;{\rm{and}}\;{d_1},{d_2}, \cdots ,{d_n} $

are pairwise coprime. Since

$ \sum\limits_{i = 1}^n {\frac{{{k_i}{m_1} \cdots {m_n}}}{{{m_i}}}} - k{m_1} \cdots {m_n} = \frac{{{m_1} \cdots {m_n}}}{M}\left( {\sum\limits_{i = 1}^n {\frac{{{k_i}M}}{{{m_i}}}} - kM} \right), $

thus we can deduce that the condition

$ \gcd \left( {\sum\limits_{i = 1}^n {\frac{{{k_i}{m_1} \cdots {m_n}}}{{{m_i}}}} - k{m_1} \cdots {m_n},q - 1} \right) = 1 $

is equivalent to $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{k}_{i}}M}{{{m}_{i}}}-kM, q-1} \right)=1$ and $\gcd \left( \frac{{{m}_{1}}\cdots {{m}_{n}}}{M}, q-1 \right)=1$. From lemma 1, one can deduce that M·gcd(m_i, m_j)|m₁…m_n. Since gcd(d_i, d_j)|(q-1) and gcd(d_i, d_j)|gcd (m_i, m_j), thus one can easily deduce that $\gcd \left( \frac{{{m}_{1}}\cdots {{m}_{n}}}{M}, q-1 \right)=1$ if and only if the integers d₁, d₂, …, d_n are pairwise coprime. This completes the proof of the claim.

In the following, we use Nq (resp. Ñq) to denote the number of the solutions of the equations (a1x1m1+a2x2m2+…+anxnmn)k=cx1k1x2k2…xtkt with xkn+1n+1…xtkt=0 (resp. xkn+1n+1…xtkt≠0). Clearly, one has

$ {N_q} = {{\bar N}_q} + {{\tilde N}_q}. $

(3)

Then, we can solve the problem in two cases. One is xkn+1n+1…xtkt=0 and the other one is xkn+1n+1…xtkt≠0.

Case ⅰ   Clearly, xkn+1n+1…xtkt=0 is reduced to say that xn+1…xt=0. Then, we have

$ \begin{array}{l} {N_q}\left[ {{x_{n + 1}} \cdots {x_t} = 0} \right] = \sum\limits_{j = 1}^{t - n} {\left( {\begin{array}{*{20}{c}} {t - n}\\ j \end{array}} \right){{\left( {q - 1} \right)}^{t - n - j}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{q^{t - n}} - {\left( {q - 1} \right)^{t - n}}. \end{array} $

(4)

Using the assumption d₁, d₂, …, d_n are pairwise coprime, it follows from (4) and lemma 4 that

$ \begin{array}{l} {{\bar N}_q} = \left( {{q^{t - n}} - {{\left( {q - 1} \right)}^{t - n}}} \right) \times \\ \;\;\;\;\;\;\;\;{N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = 0} \right] = \\ \;\;\;\;\;\;\;\;\left( {{q^{t - n}} - {{\left( {q - 1} \right)}^{t - n}}} \right) \times \\ \;\;\;\;\;\;\;\;{N_q}\left[ {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = 0} \right] = \\ \;\;\;\;\;\;\;\;{q^{n - 1}}\left( {{q^{t - n}} - {{\left( {q - 1} \right)}^{t - n}}} \right) = \\ \;\;\;\;\;\;\;\;{q^{t - 1}} - {q^{n - 1}}{\left( {q - 1} \right)^{t - n}}. \end{array} $

(5)

Case ⅱ If xkn+1n+1…xtkt≠0, we can let f=cxkn+1n+1…xtkt and d=gcd(kn+1, …, kt, q-1).

Define S:={a∈$\mathbb{F}$_q^*:ac^-1 be a d-th power element in $\mathbb{F}$_q^*}. From lemma 3, we can deduce that

$ \begin{array}{l} {{\tilde N}_q} = {N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = fx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = \\ \;\;\;\;\;\;\;\;\;d{\left( {q - 1} \right)^{t - n - 1}} \times \\ \;\;\;\;\;\;\;\;\;\sum\limits_{a \in S} {{N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = } \right.} \\ \;\;\;\;\;\;\;\;\;\left. {ax_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right]. \end{array} $

(6)

Noting that $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{k}_{i}}{{m}_{1}}\cdots {{m}_{n}}}{{{m}_{i}}}-kM, q-1} \right)=1$ and d₁, d₂, …, d_n are pairwise coprime. Thus, for any given a∈S, from the claim and lemma 5, one has

$ \begin{array}{l} {N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = } \right.\\ \;\;\;\;\;\;\;\left. {ax_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = {q^{n - 1}} + {\left( { - 1} \right)^{n - 1}}. \end{array} $

Then, it follows from lemma 6 that

$ \begin{array}{l} \sum\limits_{a \in S} {{N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = } \right.} \\ \;\;\;\;\;\;\;\left. {cx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = \frac{{q - 1}}{d}\left( {{q^{n - 1}} + {{\left( { - 1} \right)}^{n - 1}}} \right). \end{array} $

(7)

Using (6) and (7), one can derive that

$ {{\tilde N}_q} = {\left( {q - 1} \right)^{t - n}}\left( {{q^{n - 1}} + {{\left( { - 1} \right)}^{n - 1}}} \right). $

(8)

The desired result can follow immediately from (3), (5) and (8). This ends the proof of theorem 1.

In concluding this section, we present some trivial corollaries.

Corollary 1   For the further generalized Markoff-Hurwitz-type equations of the form

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = c{x_1}{x_2} \cdots {x_t} $

over the finite field $\mathbb{F}$_q, where n≥2, m_i, k and t≥n are positive integers, a_i, c∈$\mathbb{F}$_q^*, for 1≤i≤n. If $\gcd \left( \sum\limits_{i=1}^{n}{\frac{M}{{{m}_{i}}}-kM, q-1} \right)=1$ and d₁, d₂, …, d_n are pairwise coprime, then

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Corollary 2   For the further generalized Markoff-Hurwitz-type equations of the form

$ {{\left( {{a}_{1}}{{x}_{1}}^{{{m}_{1}}}+{{a}_{2}}{{x}_{2}}^{{{m}_{2}}}+\cdots +{{a}_{n}}{{x}_{n}}^{{{m}_{n}}} \right)}^{k}}=c{{x}_{1}}^{{{m}_{1}}}{{x}_{2}}^{{{m}_{2}}}\cdots {{x}_{t}}^{{{m}_{t}}} $

over the finite field $\mathbb{F}$_q, where n≥2, m_i, k and t≥n are positive integers, a_i, c∈$\mathbb{F}$_q^* for 1≤i≤n. If gcd((n-k)M, q-1)=1 and d₁, d₂, …, d_n are pairwise coprime, then

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Corollary 3   For the further generalized Markoff-Hurwitz-type equations of the form

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = c{x_1}{x_2} \cdots {x_t} $

over the finite field $\mathbb{F}$_q, where n≥2, m_i, k and t≥n are positive integers, a_i, c∈$\mathbb{F}$_q^*, for 1≤i≤n. If gcd((n-km)m, q-1)=1, then

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Clearly, corollaries 1~3 are some special cases of theorem 1. For example, consider the further generalized Markoff-Hurwitz-type equation over

$ {\mathbb{F}_{13}}:\left( {{a_1}x_1^5 + {a_2}x_2^3 + {a_3}x_3^8} \right) = bx_1^7x_2^7x_3^7x_4^{10}x_5^5x_6^4, $

(8)

where a₁, a₂, a₃, b∈$\mathbb{F}$₁₃^*. Since gcd(5, 12)=1, gcd(3, 12)=3, gcd(8, 12)=4 and gcd(7(24+40+15)-240, 12)=1 are co-prime, one can immediately conclude that (8) has 373021 solutions in $\mathbb{F}$₁₃⁶ by theorem 1.