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Fix a dimension $n \geq 1$ and a number $k \geq 1$ and suppose that $a_1,...,a_k \in \mathbb{R}^n$ are points in $n$-dimensional space such that among all the $nk$ coordinates there is a subset of size at least $nk - \binom{n+1}{2}$ that is algebraically independent (over $\mathbb{Q}$).

Is there an isometry $T$ of $\mathbb{R}^n$ such that the set of the coordinates of $Ta_1,...,Ta_k$ is algebraically independent? In other words, can we "erase" the algebraic dependency using isometries? Note that the group of isometries acting on $\mathbb{R}^n$ has $\binom{n+1}{2}$ degrees of freedom, that is, we can prescribe the position of $\binom{n+1}{2}$ coordinates, so intuitively the conditions are the weakest such that one can still hope that the answer is positive.

This is easy for $n=1$, and in general, it is easy to see how translations affect algebraic dependency. This paper has an explicit construction for $n=2$ (Lemma 3.5), but it seems difficult to generalize this to arbitrary dimension (or maybe it's just that my knowledge of the Euclidean group is lacking). I wonder if there's a high-level approach to this, or indeed, if the interplay between the 'algebraic' and 'geometric' structures on $\mathbb{R}^n$, given by algebraic independence and isometries, respectively, has been studied before.

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The answer is no.

A rational edge length is an algebraic dependence between the coordinates that you cannot get rid of by an isometry. One can find $k$ points in $\mathbb{R}^n$ with all but one coordinate algebraically independent and $\sum_{i=1}^n (a_1^i - a_2^i)^2 = 1$. (For this, take $k$ generic points with $a_1$ and $a_2$ at distance less than $1$, and then change one of the coordinates of $a_2$ so that the distance from $a_1$ becomes $1$.)

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  • $\begingroup$ I see now that I posed the question in too general terms; of course one must also require some kind of algebraic independency assumption on the edge lengths. The simplest may be to generalize the condition of the cited lemma and assume that the $i$-th point for $i \leq n$ has zeroes in the first $n+1-i$ coordinates, and that the other $nk - \binom{n+1}{2}$ coordinates are independent. Is it alright if I edit the question? Or should I ask it separately? $\endgroup$
    – Dániel G.
    Commented Nov 9, 2018 at 11:45
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    $\begingroup$ I believe, an edit should not change the question but add extra information or clarifications, see mathoverflow.net/help/editing. $\endgroup$
    – Ivan Izmestiev
    Commented Nov 9, 2018 at 11:58
  • $\begingroup$ Although there is some discussion about trivial answers to questions that are trivial because an assumption was forgotten; see e.g. here. It's not entirely clear that this is the case here, but it's something to keep in mind. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 9, 2018 at 13:37
  • $\begingroup$ Actually if the original poster would like to delete this question and to ask a new question, I have nothing against this. $\endgroup$
    – Ivan Izmestiev
    Commented Nov 9, 2018 at 14:13
  • $\begingroup$ Based on the linked meta discussion I chose to accept this answer. Since then I've made progress on the "intended" question, so there might be no need to ask it separately. $\endgroup$
    – Dániel G.
    Commented Nov 10, 2018 at 21:14

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