07-20-2021, 10:12 PM

(06-10-2018 07:15 AM)Dieter Wrote: [ -> ]So it can't be the discriminant of the original cubic equation (here D<0 means three real roots).

But it looks like it's indeed some discriminant of a reduced quadratic.

It is interesting cubic discriminant sign test is opposite of quadratic (cubic Δ < 0 → 3 real and unequal roots)

We can show the reason with identity:

x³ + y³ + z³ − 3xyz = (x+y+z) * (x+yω+z/ω) * (x+y/ω+zω), where ω = e^(i*2*pi/3)

LHS is a depressed cubic: x³ + p*x + q , where p = -3yz, q = y³ + z³

We then setup a quadratic of t, with roots (y³, z³)

(t - y³)*(t - z³) = t² - q*t - (p/3)³ → t = q/2 ± √Δ, where Δ = (q/2)² + (p/3)³

Defined this way, cubic discriminant is really quadratic discriminant (*)

If Δ < 0, t is complex, (y,z) is conjugate of each other, y + z = y + conj(y) = 2*Re(y)

All 3 roots are thus real and unequal: (y = ³√t , z = (-p/3)/y)

Δ < 0: x³ + p*x + q = (x + 2*Re(y)) * (x + 2*Re(yω)) * (x + 2*Re(y/ω))

(*) CRC Handbook of Mathematical Science 6th ed. (Beyer) also define cubic Δ this way.

But, Δ is likely defined scaled by -108, to -(4p³+27q²), matching General formula of discriminant