My pal Dan Christensen has created a fascinating picture of all the roots of all polynomials of degree £ 5 with integer coefficients ranging from -4 to 4:
Roots of quadratic polynomials are in grey; roots of cubics are in cyan; roots of quartics are in red and roots of quintics are in black. The horizontal axis of symmetry is the real axis; the vertical axis of symmetry is the imaginary axis. The big hole in the middle is centered at 0; the next biggest holes are at ±1, and there are also holes at ±i and all the sixth roots of 1.
You can see lots of fascinating patterns here, like how the roots of polynomials with integer coefficients tend to avoid integers and roots of unity - except when they land right on these points! You can see more patterns if you zoom in:
Now you see beautiful feathers surrounding the blank area around the point 1 on the real axis, a hexagonal star around exp(i π / 6), a strange red curve from this point to 1, smaller stars around other points, and more....
People should study this sort of thing! Let's define the Christensen set Cd,n to be the set of all roots of all polynomials of degree d with integer coefficients ranging from -n to n. Clearly Cd,n gets bigger as we make either d or n bigger, and it becomes dense in the complex plane as n approaches ∞, as long as d ≥ 1. We get all the rational complex numbers if we fix d ≥ 1 and let n → ∞, and all the algebraic complex numbers if let both d,n → ∞. Based on the above picture, there seem to be lots of interesting conjectures to make about what it does as d → ∞, for fixed n.
For more on this sort of thing, see:
Odlyzko and Poonen proved some interesting things about the set of all roots of all polynomials with coefficients 0 or 1. If we define a fancier Christensen set Cd,p,q to be the set of roots of all polynomials of degree d with coefficients ranging from p to q, Odlyzko and Poonen are studying Cd,0,1 in the limit d → ∞. They mention some known results and prove some new ones: this set is contained in the half-plane Re(z) < 3/2 and contained in the annulus 1/Φ < |z| < Φ where Φ is the golden ratio (√5 + 1)/2. In fact they trap it, not just between these circles, but between two subtler curves. They also show that the closure of this set is path connected but not simply connected.