1
$\begingroup$

Suppose an algebraic variety over Q, or subvariety of one, has a parametrization, also over Q.

Clearly an infinite number of birationally equivalent parametrisations can be obtained from this. But some will be better than others in respect of small-height rational points being consistently obtainable from small-height values of all the parameters.

I wondered if there is a name for this property, such as (off the top of my head) "well conditioned", or "balanced", or something along those lines, and maybe even a quantitative theory (an aspect of Diophantine Approximation perhaps) of how well a parametric solution can have this "balanced" property for all the rational points it includes.

For some higher-dimensional [sub]varieties, I imagine it may be that no single parametric solution is optimal in this sense for all rational points, any more than (using a metric metaphor) no single point is "near" all the others.

edit: Following added for clarification:

As an example, the surface defined by $x^2 + y^2 - z^2 = 1$ has a simple rational parametrization $x = (u v - 1)/(u - v)$, $y = (u + v)/(u - v)$, $z = (u v + 1)/(u - v)$, where $u, v$ are two parameters (which can each be expressed in terms of "x, y, z").

Obviously, these parameters can be replaced by any other pair related to them by an invertable affine transform with rational coefficients (or more generally by birational transforms, but an affine transform is sufficient as an example).

For example, letting $u = 1000 p + q$, $v = 999999 p + 1000 q$ gives an equally valid general parametrization in terms of $p, q$; but clearly this is less suited to obtaining small-height rational points (x, y, z) for small-height rational parameter values, because the "lopsided" transform throws the parametrization out of whack in that respect.

I agree that the concept, as far as have been able to sketch it, may seem somewhat wooly, and possibly not very useful. But I believe a notion of entropy can be defined for sets of birational transforms. So perhaps that and related ideas may have some bearing on the question.

$\endgroup$
1
  • 1
    $\begingroup$ Can you give some examples to illustrate the kind of phenomenon you are interested in? At the moment the question is very vague. I think it seems doubtful there could be a reasonable answer - problems about the height of the smallest rational point or counting rational points of bounded height can be incredibly difficult, even for rational varieties. A prime example being cubic surfaces (yes, even the rational cubic surfaces are incredibly difficult for such problems). $\endgroup$ Feb 11, 2016 at 21:23

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.