discriminants

Polynomial discriminant is an expression based on the polynomial coefficients that vanishes if and only if the polynomial has multiple roots. Discriminants are often denoted as D or Δ. For example for the quadratic polynomial ax²+bx+c=0 its discriminant is a well known Δ=b²-4ac. For precise definition and more details see for example Wikipedia or Wolfram MathWorld. Mentioned Wikipedia article contains the following information:

"For higher degrees, the discriminant is always a polynomial function of the coefficients. It becomes significantly longer for the higher degrees. The discriminant of a general quartic has 16 terms, that of a quintic has 59 terms, that of a 6th degree polynomial has 246 terms, and the number of terms increases exponentially with the degree."

I decided to look at this more closely and to get explicit formulas for higher degrees. See the following results:

See also the attached file with explicit formulas.

The results suggest that indeed the number of terms may increase exponentially. Of course it is nearly impossible to deduce exact asymptotic behavior having only 10 first elements of the infinite sequence.