View source for Hurwitz equation

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''Markoff–Hurwitz equation, Markov–Hurwitz equation''

A Diophantine equation (cf. [[Diophantine equations|Diophantine equations]]) of the form

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

for fixed $a,n\in\mathbf Z^+$, $n\geq3$. The case $n=a=3$ was studied by A.A. Markoff [A.A. Markov] [[#References|[a1]]] because of its relation to [[Diophantine approximations|Diophantine approximations]] (cf. also [[Markov spectrum problem|Markov spectrum problem]]). More generally, these equations were studied by A. Hurwitz [[#References|[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 $\{x_1,\dotsc,x_n\}$, 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 \ref{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 \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 $(1,\dotsc,1)$ together with the trivial solution. N.P. Herzberg [[#References|[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 [[#References|[a4]]] showed that for any $r$ there exists a pair $(a,n)$ such that \ref{a1} has at least $r$ fundamental solutions.

D. Zagier [[#References|[a5]]] investigated the asymptotic growth for the number of solutions to the Markov equation ($a=n=3$) below a given bound, and Baragar [[#References|[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 [[#References|[a7]]] and G. Rosenberger [[#References|[a8]]]. L. Wang [[#References|[a9]]] studied a class of smooth variations.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Markoff,   "Sur les formes binaires indéfinies"  ''Math. Ann.'' , '''17'''  (1880)  pp. 379–399</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.P. Herzberg,   "On a problem of Hurwitz"  ''Pacific J. Math.'' , '''50'''  (1974)  pp. 485–493</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Baragar,   "Integral solutions of Markoff–Hurwitz equations"  ''J. Number Th.'' , '''49''' :  1  (1994)  pp. 27–44</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Zagier,   "On the number of Markoff numbers below a given bound"  ''Math. Comp.'' , '''39'''  (1982)  pp. 709–723</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Baragar,   "Asymptotic growth of Markoff–Hurwitz numbers"  ''Compositio Math.'' , '''94'''  (1994)  pp. 1–18</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G. Rosenberger,   "Über die Diophantische Gleichung $ax^2+by^2+cz^2=dxyz$"  ''J. Reine Angew. Math.'' , '''305'''  (1979)  pp. 122–125</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  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</TD></TR></table>