Squares inscribed on algebraic plane curves

Toeplitz conjectured in 1911 that every Jordan curve contains the
four corners of some square. The past century has seen partial
results, but the full conjecture remains open.

For my master's thesis I considered algebraic plane curves instead
of Jordan curves. I will present examples of inscribed squares and
a sketch of the proof of the main result of my thesis, that an
algebraic plane curve of degree m inscribes either infinitely many
squares or at most (m^4 - 5m^2 + 4m)/4 squares.

The proof uses Bernshtein's theorem on the number of solutions to a
polynomial system of equations, no background in algebraic geometry
or polytope theory will be assumed.