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A Diophantine equation (cf. [[Diophantine equations|Diophantine equations]]) of the form
 
A Diophantine equation (cf. [[Diophantine equations|Diophantine equations]]) of the form
  
$$x_1^2+\ldots+x_n^2=ax_1\ldots x_n\tag{a1}$$
+
$$x_1^2+\dotsb+x_n^2=ax_1\dotsm x_n\label{a1}\tag{a1}$$
  
for fixed $a,n\in\mathbf Z^+$, $n\geq3$. The case $n=a=3$ was studied by A.A. Markoff [A.A. Markov] [[#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 $\{x_1,\ldots,x_n\}$, sign changes of pairs of variables, and the mapping
+
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 \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
  
$$\sigma(x_1,\ldots,x_n)=(ax_2\ldots x_n-x_1,x_2,\ldots,x_n).$$
+
$$\sigma(x_1,\dotsc,x_n)=(ax_2\dotsm x_n-x_1,x_2,\ldots,x_n).$$
  
If \ref{a1} has an integer solution $P$ and $P$ is not the trivial solution $(0,\ldots,0)$, 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 [[#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 $\{P_1,\ldots,P_r\}$ 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 [[#References|[a4]]] showed that for any $r$ there exists a pair $(a,n)$ such that \ref{a1} has at least $r$ fundamental solutions.
+
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 [[#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 \eqref{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 $n\geq4$.
 
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 $n\geq4$.
Line 17: Line 17:
  
 
====References====
 
====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>
+
<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 {{ZBL|12.0143.02}}</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 {{ZBL|0247.10010}}</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>

Latest revision as of 16:51, 14 August 2023

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 Zbl 0247.10010
[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
How to Cite This Entry:
Hurwitz equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_equation&oldid=32900
This article was adapted from an original article by A. Baragar (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article