Homogeneous Interpolation and Some Continued Fractions

TL;DR

This paper establishes bounds on the existence of algebraic curves passing through general points with specified multiplicities, using continued fractions to express these bounds, advancing understanding in algebraic geometry.

Contribution

It introduces bounds for curves passing through points with multiplicities, expressed via palindromic continued fractions, extending previous results in algebraic geometry.

Findings

No curve of degree d passing through 10 points with multiplicity m exists if d/m < 2280/721.

Bounds are expressed as palindromic continued fractions.

Results apply to other specific point configurations.

Abstract

We prove: if $d/m < 2280/721$, there is no curve of degree $d$ passing through $n = 10$ general points with multiplicity $m$ in $\bf{P}^2$. Similar results are given for other special values of $n$. Our bounds can be naturally written as certain palindromic continued fractions.