I assume this is one of these things where it is just like ... Oh I don't know this kind of math.
Because in my mind I take a "random" polynomial and only consider the largest two roots. Then I consider doing two reflections: across the bottom/top of the curve, and across the x axis. If those top two roots aren't degenerate the combination of the four curves using the reflections have two curves with a largest real root, and two curves with a largest imaginary root (I think). If it is degenerate then the largest root is real.
So I would then conclude that (1) there are more real largest roots than imaginary and (2) the "advantage" is vanishingly small since there are infinitely more curves with a largest non-degenerate root than degenerate root.
As one can tell I barely know the proper nomenclature. I assume I'm mainly missing some consideration about the uniformity of random coefficients not resulting in a uniform distribution in space? Or possibly my reflections can violate one of the conditions (real coefficients?)? Or I'm just plain wrong.
I don't follow your reflections. Reflecting the graph of a polynomial p(x) along the x-axis means replacing p(x) by p(-x). What do you mean by reflecting "across the bottom/top of the curve"? Reflecting along the y-axis? This would mean p(x) is replaced by -p(x).
Ah there we go. At the very least I was only considering the subspace where the polynomial's derivative has only real roots?
Think of a simple quadratic. The reflection is across the min(f(x)) axis. But for higher order polynomials that doesn't always work so that is certainly an issue. Thanks
Because in my mind I take a "random" polynomial and only consider the largest two roots. Then I consider doing two reflections: across the bottom/top of the curve, and across the x axis. If those top two roots aren't degenerate the combination of the four curves using the reflections have two curves with a largest real root, and two curves with a largest imaginary root (I think). If it is degenerate then the largest root is real.
So I would then conclude that (1) there are more real largest roots than imaginary and (2) the "advantage" is vanishingly small since there are infinitely more curves with a largest non-degenerate root than degenerate root.
As one can tell I barely know the proper nomenclature. I assume I'm mainly missing some consideration about the uniformity of random coefficients not resulting in a uniform distribution in space? Or possibly my reflections can violate one of the conditions (real coefficients?)? Or I'm just plain wrong.