Simultaneous Arithmetic Progressions on Algebraic Curves

TL;DR

This paper establishes an upper bound of 4319 on the length of simultaneous arithmetic progressions on real algebraic curves, using graph crossing bounds, Bezout's Theorem, and convex curve properties, and extends results to complex curves.

Contribution

It provides the first explicit bound on s.a.p. length on algebraic curves over R and extends the analysis to complex curves, combining geometric and algebraic methods.

Findings

Bound of 4319 on s.a.p. length for curves over R.

Method using graph crossings and Bezout's Theorem.

Extension of results to complex algebraic curves.

Abstract

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.