Wikipedia:Reference desk/Archives/Mathematics/2018 December 8

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December 8

AGM

I was reading the article Arithmetic–geometric mean (AGM). I was wondering, is the AGM of two algebraic numbers necessarily transcendental? More generally, when is it transcendental and when is it not? I was hoping to get such information added to the AGM article.--Solomonfromfinland (talk) 06:13, 8 December 2018 (UTC)

@Solomonfromfinland: Here’s how to find an infinitude of cases with algebraic (non-transcendental) AGM. Consider hypothesized initial conditions $a_{0},g_{0}$ that converge exactly, rather than asymptotically, to AGM. The initial conditions $a_{0},g_{0}$ are our unknowns, and we require that the system converge exactly to $a_{2}=g_{2}.$ Use the given iterative formula twice to obtain

$a_{2}={\frac {a_{0}+g_{0}}{4}}+{\frac {\sqrt {a_{0}g_{0}}}{2}},$

$g_{2}={\sqrt {{\frac {a_{0}+g_{0}}{2}}{\sqrt {a_{0}g_{0}}}}}.$

Then equate $a_{2}=g_{2}$, square both sides, rearrange, and isolate the remaining square root and square both sides. The result is a polynomial equation in $a_{0}$ and $g_{0}.$ For any arbitrary algebraic value of $g_{0},$ each of the solutions for $a_{0}$ is an algebraic number. Then by the two equations above, $a_{2}=g_{2}=AGM$ is an algebraic number.

This approach presumably works for any finite number of steps to exact convergence. We can still hypothesize that all cases that converge only asymptotically converge to a trancendental number. Loraof (talk) 00:12, 10 December 2018 (UTC)

It's not hard to show that a_i+1=g_i+1 implies a_i=g_i, so you can't get a_n=g_n unless you start with a₀=g₀. Technically the answer to the original question is no since if you start with both numbers as the same algebraic number then you get the AGM is algebraic. But presumably the word 'distinct' was meant to be in the question somewhere. --RDBury (talk) 06:28, 10 December 2018 (UTC)

I disagree with “so you can't get a_n=g_n unless you start with a₀=g₀“. In nonlinear difference equations, it’s quite possible to have more than one stationary point after two iterations even when there is a one-iteration stationary point. The one-iteration stationary point is just one of the two-period stationary points. Loraof (talk) 17:36, 10 December 2018 (UTC)

I stand corrected—you are right, a0=g0 is the only possibility for this to work. Loraof (talk) 21:05, 10 December 2018 (UTC)