I gave this to John Cremona who has an algorithm for finding a point (if it exists) on a diagonal conic over a function field. The equation in question is an example of such -- if we clear denominators we get: (2-q)^2*h^2 = (2-q^2)*g^2 + 1/2*(2-q)^2*(2-q^2) (*) So this is a conic in (g,h) over the field Q(q). It's well know that if you have one solution to a conic, all the others are generated by a rational parametrization from that one. However, John's program showed that (*) had no points in Q(q). Warut's parametrized solutions essentially amount to finding one point on (*) in a quadratic extension of Q(q) (his parameter, r, satisfies a quadratic over Q(q)), and then using the general recipe to generate all other solutions over that field. An alternative approach would be to look at (*) as an elliptic curve in (h,q) over Q(g), which must have positive rank since we know that there are an infinite number of solutions. Victor On Tue, Sep 1, 2009 at 11:13 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
It's always nice to have examples of geometrical configurations with rational parameters --- apart from anything else, they're useful for testing software.
A polytore with square section edge 2, outer cuboid depth g, inner quadrangle height h, inner radius q, is planar just when
h^2/(2-q^2) = g^2/(2-q)^2 + 1/2
but this equation appears to have no small rational solutions, at any rate when 0 < q < 1.
Is it soluble over the rationals? WFL
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