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Rational points near curves and small nonzero |x^3-y^2| via lattice reduction

Noam D. Elkies

TL;DR

The paper introduces a lattice-reduction framework to efficiently locate rational points near plane curves by partitioning the curve into short linear patches and reducing the nearby lattice points in small parallelepipeds, achieving space complexity $O(\log N)$ and heuristic time $(N+\delta N^3)\log^{O(1)} N$; the method extends to small nonzero values of $|x^3-y^2|$ with a rigorous $O(X^{1/2}\log^{O(1)} X)$ bound. It applies the approach to Fermat curves, including the cubic, yielding new numerical data and insights into polynomial families that generate near-misses, and demonstrates the method’s versatility across algebraic and nonalgebraic curves. The Hall conjecture is tackled by a specialized 2D lattice-reduction variant, delivering a rigorous bound and substantial computational results up to $X=10^{18}$, including new records for $x^{1/2}/|x^3-y^2|$. Overall, the work provides a practical, parallelizable algorithm with strong theoretical underpinnings that advances both computational capabilities and conceptual understanding of rational points near curves and related Diophantine problems.

Abstract

We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat cubic, we find all integer solutions of |x^3+y^3-z^3|<M with 0<x<=y<z<N in heuristic time << L(N) M [where L(X):= (log X)^O(1)] provided M>>N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M<N^3), the computational costs are essentially as low as possible. Moreover the algorithm readily parallelizes. It not only yields new numerical examples but leads to theoretical results, difficult open questions, and natural generalizations. We also adapt our algorithm to investigate Hall's conjecture: we find all integer solutions of 0<|x^3-y^2|<<x^(1/2) with x<X in time O(X^(1/2) L(X)). By implementing this algorithm with X=10^18 we shattered the previous record for x^(1/2)/|x^3-y^2|. The O(X^{1/2} L(X)) bound is rigorous; its proof also yields new estimates on the distribution mod 1 of sqrt(cx^3) for any positive rational c.

Rational points near curves and small nonzero |x^3-y^2| via lattice reduction

TL;DR

The paper introduces a lattice-reduction framework to efficiently locate rational points near plane curves by partitioning the curve into short linear patches and reducing the nearby lattice points in small parallelepipeds, achieving space complexity and heuristic time ; the method extends to small nonzero values of with a rigorous bound. It applies the approach to Fermat curves, including the cubic, yielding new numerical data and insights into polynomial families that generate near-misses, and demonstrates the method’s versatility across algebraic and nonalgebraic curves. The Hall conjecture is tackled by a specialized 2D lattice-reduction variant, delivering a rigorous bound and substantial computational results up to , including new records for . Overall, the work provides a practical, parallelizable algorithm with strong theoretical underpinnings that advances both computational capabilities and conceptual understanding of rational points near curves and related Diophantine problems.

Abstract

We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat cubic, we find all integer solutions of |x^3+y^3-z^3|<M with 0<x<=y<z<N in heuristic time << L(N) M [where L(X):= (log X)^O(1)] provided M>>N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M<N^3), the computational costs are essentially as low as possible. Moreover the algorithm readily parallelizes. It not only yields new numerical examples but leads to theoretical results, difficult open questions, and natural generalizations. We also adapt our algorithm to investigate Hall's conjecture: we find all integer solutions of 0<|x^3-y^2|<<x^(1/2) with x<X in time O(X^(1/2) L(X)). By implementing this algorithm with X=10^18 we shattered the previous record for x^(1/2)/|x^3-y^2|. The O(X^{1/2} L(X)) bound is rigorous; its proof also yields new estimates on the distribution mod 1 of sqrt(cx^3) for any positive rational c.

Paper Structure

This paper contains 15 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The algorithm in theory
  4. Specification and heuristic analysis
  5. Can the estimates be made rigorous?
  6. Variations and generalizations
  7. Rational points on varieties
  8. The algorithm in practice
  9. Fermat curves of degree $>3$
  10. The Fermat cubic
  11. Miscellaneous examples
  12. Hall's conjecture
  13. Review of Hall's conjecture
  14. The new algorithm
  15. Numerical results

Key Result

lemma 1

For each positive integer $n$ and positive real $t$ there exists an effective constant $M_n(t)$ such that the following bound holds: for any lattice $\Lambda\subset{\bf R}^n$ whose dual lattice $\Lambda^*$ has no nonzero vector of length $<r$, and for any $R>0$, there are at most $M_n(rR) \, r^{-n}

Theorems & Definitions (14)

  • lemma 1
  • proof
  • corollary 1
  • proof
  • theorem 1
  • theorem 2
  • proof
  • theorem 3
  • theorem 4
  • proof
  • ...and 4 more