Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 4 / c / ii

Table of contents
- Solution ii

Solution

Replacing

[2^{r}]

and

4^{r}

by

[3^{r}]

and

9^{r}

gives the same canonical height. Indeed

h ([3] P) = 9 h (P) + O_{E} (1)

, and the same telescoping argument constructs a quadratic height differing from the original by a bounded function. A bounded quadratic function is zero, so the two limits agree.

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