Hurwitz equation

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Markoff–Hurwitz equation, Markov–Hurwitz equation

A Diophantine equation (cf. Diophantine equations) of the form

$$x_1^2+\ldots+x_n^2=ax_1\ldots x_n\tag{a1}$$

for fixed $a,n\in\mathbf Z^+$, $n\geq3$. The case $n=a=3$ was studied by A.A. Markoff [A.A. Markov] [a1] because of its relation to Diophantine approximations (cf. also Markov spectrum problem). More generally, these equations were studied by A. Hurwitz [a2]. These equations are of interest because the set of integer solutions to \ref{a1} is closed under the action of the group of automorphisms $\mathcal A$ generated by the permutations of the variables $\{x_1,\ldots,x_n\}$, sign changes of pairs of variables, and the mapping

$$\sigma(x_1,\ldots,x_n)=(ax_2\ldots x_n-x_1,x_2,\ldots,x_n).$$

If \ref{a1} has an integer solution $P$ and $P$ is not the trivial solution $(0,\ldots,0)$, then its $\mathcal A$-orbit $\mathcal A(P)$ is infinite. Hurwitz showed that if \ref{a1} has a non-trivial integer solution, then $a\leq n$; and if $a=n$, then the full set of integer solutions is the $\mathcal A$-orbit of $(1,\ldots,1)$ together with the trivial solution. N.P. Herzberg [a3] gave an efficient algorithm to find pairs $(a,n)$ for which the Hurwitz equation has a non-trivial solution. Hurwitz also showed that for any pair $(a,n)$ there exists a finite set of fundamental solutions $\{P_1,\ldots,P_r\}$ such that the orbits $\mathcal A(P_i)$ are distinct and the set of non-trivial integer solutions is exactly the union of these orbits. A. Baragar [a4] showed that for any $r$ there exists a pair $(a,n)$ such that \ref{a1} has at least $r$ fundamental solutions.

D. Zagier [a5] investigated the asymptotic growth for the number of solutions to the Markov equation ($a=n=3$) below a given bound, and Baragar [a6] investigated the cases $n\geq4$.

There are a few variations to the Hurwitz equations which admit a similar group of automorphisms. These include variations studied by L.J. Mordell [a7] and G. Rosenberger [a8]. L. Wang [a9] studied a class of smooth variations.