Matematické Fórum

Skrytý text:

For n odd no such condition exists.
For n even such condition can be: an>0, ai=0 (i<>n).
But I guess you want to formulate equivalent condition, ie. such that the equation has not real roots if and only if some condition (to be found) holds, am I right?
I also suppose that ai are real numbers, aren't?

Skrytý text:

As we said, we can only explore the case when n is even (n=2k). In this case we can rewrite left side of an equation as k products of the form . The roots of the equations are all "unreal" if and only if all are irreducible which is if and only if a discriminant of is negative for all i. So we can then formulate the condition:

The solution of the above equation has no real roots if and only if:
There exist real numbers ,, i=1..k such that:
1)
2) is the coeffitient of in (which can be expressed by and ).