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  "body": "\n## Summary\n\nLet $\\phi$ be a morphism of degree $d \\ge 2$ of projective space over a number field.\nIts canonical height is the [[height function|F1FF505PWH]] adapted to $\\phi$: it multiplies by exactly $d$ under $\\phi$.\nIt differs from the naive Weil height by a bounded amount, and it vanishes exactly at the preperiodic points of $\\phi$.\nFor the multiplication maps on an elliptic curve the same construction gives the Néron–Tate height, a quadratic form on the group of rational points.\nCanonical heights are a basic tool of [[arithmetic dynamics|BE4X0QG8XW]].\n\n## Explanations\n\n### L1 human\n\nThe height of a fraction measures how many digits it needs.\nApplying a map like $z \\mapsto z^2 + 1$ to a fraction roughly doubles the number of digits, but not exactly.\nThe canonical height corrects this: it is a slightly adjusted size for which applying the map doubles the size exactly.\nPoints of canonical height zero are those whose orbit never grows, which are exactly the points that eventually cycle.\n",
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