Curves and Tangents
In algebraic curves class on Friday, Joe Harris mentioned that the problem of whether there exist curves where any tangent intersects the curve at another point is open. This seemed like an interesting problem so I did some thinking about it. I realized that the case where one of these "second intersections" is really the point of tangency is probably a more doable case since you don't need to worry whether the points on the tangent lines you are getting are really part of the same curve. Anyway, restricting to C and doing some analysis, I managed to show that there exists a counter-example if and only if two polynomials of the form:
d/dx [(x^n-1)/(x-1)]
have a common root. Computer experiment shows that these polynomials are all irreducible, but I don't know how to prove it. If I could though, it would give me a perhaps novel proof of the conjecture in the genus 0 case, and provide some leverage on the genus 1 case.
d/dx [(x^n-1)/(x-1)]
have a common root. Computer experiment shows that these polynomials are all irreducible, but I don't know how to prove it. If I could though, it would give me a perhaps novel proof of the conjecture in the genus 0 case, and provide some leverage on the genus 1 case.