A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates.

A result (Lemma 3.3) from "Globally linked pairs of vertices in equivalent realisations of graphs" by Jackson, Jordan and Szabadka, shows that, modulo a minor detail about isometries, algebraic independence of a set of $n$ points is equivalent to algebraic independence of a set* of $2n-3$ Euclidean distances between those points.

*clearly you can't choose any such set, but there is a quite natural condition you must satisfy.

I am wondering if it is possible to say something similar for sets of points in a real 2-dimensional normed space? If it makes things easier, I think the case of $\mathbb{R}^2$ equipped with an $l_p$ norm, $p\neq 2$, (probably you want to exclude the $1$ and $\infty$ norms), would be interesting.

Is there any literature where algebraic independence/transcendence degree is considered in a set up with a non-Euclidean norm/metric?

  • $\begingroup$ I don't see how you would be able to say anything in general (since the algebricity of the $l^2$ norm is obviously essential); the $l^p$ case is obviously quite special. $\endgroup$ – Igor Rivin Jul 2 '15 at 20:36
  • $\begingroup$ Yes, that's what I fearer. Even the case $p=4$ is not clear to me... $\endgroup$ – user62562 Jul 3 '15 at 18:02
  • $\begingroup$ By the way, I had trouble backing out the statement from the lemma you mention (I did not try very hard to unravel all the jargon, it's true...) $\endgroup$ – Igor Rivin Jul 3 '15 at 19:13
  • $\begingroup$ Hmmm, I could add more details to the description if it would help? I didn't include any of the jargon originally because the underlying thought is simple - I want a link between 'generic' arrangements of points and 'generic' sets of distances in real normed linear spaces. I guess the notion of generic would not be via algebraic independence, as you say, but it would have been much nicer for me if there was some such notion. I had hoped someone would have considered this sort of thing at least for $l_p$ norms with $p>1$ an integer. $\endgroup$ – user62562 Jul 5 '15 at 8:13

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