The algebraic square peg problem

The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the case. Hundred years later we only have partial results for curves with additional smoothness properties. The contribution of this thesis is an algebraic variant of the square peg problem. By casting the set of squares inscribed on an algebraic plane curve as a variety and applying Bernshtein's Theorem we are able to count the number of such squares. An algebraic plane curve defined by a polynomial of degree m inscribes either an infinite amount of squares, or at most (m4 - 5m2 + 4m)= 4 squares. Computations using computer algebra software lend evidence to the claim that this upper bound is sharp for generic curves. Earlier work on Toeplitz's conjecture has shown that generically an odd number of squares is inscribed on a smooth enough Jordan curve. Examples of real cubics and quartics suggest that there is a similar parity condition on the number of squares inscribed on some topological types of algebraic plane curves that are not Jordan curves. Thus we are led to conjecture that algebraic plane curves homeomorphic to the real line inscribe an even number of squares