Given a quartic polynomial $f(z)$, where $z\in\mathbb C$, is it possible to find a value for $c$ where $f(z)-c=0$ only has read roots, given that there are 2 real roots?.
In other words, the range of $c$ where the local maxima are $\geq 0$ and the local minima are $\leq 0$.
Is there a discriminant style approach to this?
Edit: I meant only real roots, and the coefficients are real.
On
Consider the quartic $f(x) = x^4 + x^2 - 2$, which has two real roots $\pm 1$. It is clear from the fact tht $f$ is increasing on $[0,\infty)$ and decreasing on $(-\infty,0]$ that $f(x) - c$ always has $0$, $1$ or $2$ real roots, and the only case where any of these are repeated is $c=-2$, where the roots are $0,0, \pm i$. There is no $c$ for which all the roots are real.
No, this is not possible. Consider only monic polynomials, since multiplying by a non-zero constant does not change the roots. A monic polynomial with only real roots necessarily has real coefficients. Thus any monic polynomial of degree $4$ with at least one non-real coefficient for either $z$ or $z^2$ doesn't satisfy the property that you want (since adding a constant complex number won't change these coefficients). To give an example with two real roots, consider $$f(z)=z^2(z-i)^2=z^4-2iz^3-z^2.$$