Discriminant Conjecture

    The discriminant is easy to compute for the problem of four lines. If the lines are tangent to the rational normal curve at the points g(s), g(t), g(u), and g(v), then the discriminant is

16 ( (s-t)2(u-v)2 + (s-u)2(t-v)2 + (s-v)2(t-u)2 ) .

    Shapiro's Conjecture was proven for m=3 and p=2 by computing a specialization of the discriminant where one parameter was set to 0 and another was set to infinity.
The discriminant has degree 20 in four parameters s1, s2, s3, and s4, with 711 terms.
Brute force techniques showed that it was a sum of 232 squares, each a product of square differences of the parameters, (si - sj )2 or squares of the parameters si2 (when sj=0).

Discriminant Conjecture
The discriminant of any polynomial system formulation of Shapiro's Conjecture is a sum of products of squared differences in the parameters si.

    This has been proven in a few cases.