Degree 7, real and imaginary parts of the coefficients are rational.
The polynomial has two simple complex roots with extremely
small real separation (33 common digits) and imaginary parts close
to zero:
The remaining roots are well separated:
Define the condition number of the simple root of the polynomial
as
, in this way
a relative perturbation of modulus in the coefficients
generates
a relative change in the root of modulus at most
.
Then, for the above pair of roots we have , that is, in order to
separate the two above roots by using a floating point computation, at
least 68 digits of working precision are needed.
More difficult situations can be obtained with smaller values of the
parameter c. For
we have a pair of roots with 113
common digits and nonzero imaginary part of modulus less than
.
For we have a pair of roots with 385
common digits and nonzero imaginary part of modulus less than
.