Hurwitz equation

Markoff–Hurwitz equation, Markov–Hurwitz equation

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

$$x_1^2+\dotsb+x_n^2=ax_1\dotsm x_n\label{a1}\tag{a1}$$

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 \eqref{a1} is closed under the action of the group of automorphisms $\mathcal A$ generated by the permutations of the variables $\{x_1,\dotsc,x_n\}$, sign changes of pairs of variables, and the mapping

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

If \eqref{a1} has an integer solution $P$ and $P$ is not the trivial solution $(0,\dotsc,0)$, then its $\mathcal A$-orbit $\mathcal A(P)$ is infinite. Hurwitz showed that if \eqref{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 $(1,\dotsc,1)$ 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 $\{P_1,\dotsc,P_r\}$ 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 \eqref{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.

References

[a1]	A.A. Markoff, "Sur les formes binaires indéfinies" Math. Ann. , 17 (1880) pp. 379–399 Zbl 12.0143.02
[a2]	A. Hurwitz, "Über eine Aufgabe der unbestimmten Analysis" Archiv. Math. Phys. , 3 (1907) pp. 185–196 (Also: Mathematisch Werke, Vol. 2, Chapt. LXX (1933 and 1962), 410–421)
[a3]	N.P. Herzberg, "On a problem of Hurwitz" Pacific J. Math. , 50 (1974) pp. 485–493
[a4]	A. Baragar, "Integral solutions of Markoff–Hurwitz equations" J. Number Th. , 49 : 1 (1994) pp. 27–44
[a5]	D. Zagier, "On the number of Markoff numbers below a given bound" Math. Comp. , 39 (1982) pp. 709–723
[a6]	A. Baragar, "Asymptotic growth of Markoff–Hurwitz numbers" Compositio Math. , 94 (1994) pp. 1–18
[a7]	L.J. Mordell, "On the integer solutions of the equation $x^2+y^2+z^2+2xyz=n$" J. London Math. Soc. , 28 (1953) pp. 500–510
[a8]	G. Rosenberger, "Über die Diophantische Gleichung $ax^2+by^2+cz^2=dxyz$" J. Reine Angew. Math. , 305 (1979) pp. 122–125
[a9]	L. Wang, "Rational points and canonical heights on K3-surfaces in $\mathbf P^1\times\mathbf P^1\times\mathbf P^1$" Contemp. Math. , 186 (1995) pp. 273–289