Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 5 / c

Solution

For

x (P) = A / B^{2}

in lowest terms, take the logarithmic naive height

h (P) = lo g max {∣ A ∣, B^{2}}

. Its required properties are

h (2 P) = 4 h (P) + O (1)

and

h (P + Q) + h (P - Q) = 2 h (P) + 2 h (Q) + O (1),

h (P) = lim_{r \to \infty} 4^{- r} h (2^{r} P) .

The first bounded-error relation makes this a convergent telescoping correction to

h (P)

. Apply the second relation to

2^{r} P, 2^{r} Q

, divide by

4^{r}

, and let

r \to \infty

to obtain

h (P + Q) + h (P - Q) = 2 h (P) + 2 h (Q) .

Also

h (2 P) = 4 h (P)

, and the parallelogram identity then gives

h (n P) = n^{2} h (P)

for every integer

n

. Thus

h

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