# Mason-Stothers theorem

Mason’s theorem is often described as the polynomial case of the (currently unproven) ABC conjecture^{}.

###### Theorem 1 (Mason-Stothers).

Let $f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{,}g\mathit{}\mathrm{(}z\mathrm{)}\mathrm{,}h\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\in}\mathrm{C}\mathit{}\mathrm{[}z\mathrm{]}$ be such that $f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{+}g\mathit{}\mathrm{(}z\mathrm{)}\mathrm{=}h\mathit{}\mathrm{(}z\mathrm{)}$ for all $z$, and such that $f$, $g$, and $h$ are pair-wise relatively prime. Denote the number of distinct roots of the product $f\mathit{}g\mathit{}h\mathit{}\mathrm{(}z\mathrm{)}$ by $N$. Then

$\mathrm{max}\mathrm{deg}\{f,g,h\}+1\le N.$ |

Title | Mason-Stothers theorem |
---|---|

Canonical name | MasonStothersTheorem |

Date of creation | 2013-03-22 13:49:24 |

Last modified on | 2013-03-22 13:49:24 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 4 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 30C15 |

Synonym | Mason’s Theorem |

Related topic | PolynomialAnalogonForFermatsLastTheorem |