Very interesting idea, Warren. Thanks! At 02:33 PM 7/1/2015, Warren D Smith wrote:
If you also want to know not merely about the absolute value, but also about the phase angle... i.e. about Im(Log(discriminant)), that's kind of stupid physically since it is not rotation invariant.... but it's sort of like the quadrupole moment of the charge configuration, but not exactly.
--Actually, a cheap statement about a=arg(discriminant) for a monic polynomial P(z) of degree N, is that exp(i*a/(N-1)) * P(z * exp(-i*a/((N-1)*N) ) ) is an "improved" version of P(z) having same degree, still monic, but now with pure-real discriminant with same old absolute value.
So it tells you how much you need to rotate in order to orient P(z) "correctly." Any pure-real polynomial P(z) already was oriented correctly without you needing to do anything.
It is intuitively somewhat like quadrupole moment as I'd said, e.g. a pure-real polynomial automatically has quadrupole moment pointing in the real (or pure imaginary) direction.