The algebraic square peg problem

TL;DR

This paper introduces an algebraic approach to the square peg problem by representing inscribed squares on algebraic curves as varieties and using Bernshtein's Theorem to count them, providing new bounds and computational evidence.

Contribution

It develops an algebraic variant of the square peg problem, deriving an explicit upper bound on the number of inscribed squares on algebraic curves using algebraic geometry techniques.

Findings

Algebraic curves inscribe either infinitely many squares or at most a specific finite number.

The derived upper bound is sharp for generic curves based on computational evidence.

The approach connects classical geometric problems with algebraic geometry methods.

Abstract

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 (m^4 - 5m^2 + 4m)/4 squares. Computations using computer algebra software lend evidence to the claim that this upper bound is sharp for generic curves.