Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 20 / 7

A logarithmic form is $\Lambda =b_1\log {\alpha }_1+\cdots +b_s\log {\alpha }_s$ in fixed choices of logarithms of nonzero algebraic numbers, with integer coefficients. The indispensable condition is $\Lambda \ne 0$. Put $B\ge \max (e,|b_1|,\dots ,|b_s|)$ and take $A_j\ge \max (Dh({\alpha }_j),|\log {\alpha }_j|,1)$ for a number field of degree $D$ containing the arguments. A typical effective estimate iswith an explicitly computable constant. For fixed arguments this is a power-type lower bound $|\Lambda |>B^{-C}$. It is fundamentally stronger for applications than a crude norm estimate exponential in $B$.

The basic estimates follow the auxiliary-function pattern already used here, in a multivariable form. Choose a polynomial or an interpolation determinant in powers of the algebraic arguments; impose many derivative or interpolation conditions by Siegel lemma. Analytic extrapolation near the chosen logarithms makes the determinant tiny. Arithmetic denominator and norm estimates prevent a nonzero algebraic determinant from being so small, while a multiplicity or zero estimate ensures the necessary determinant does not vanish identically. Balancing the degree, multiplicity and arithmetic height parameters produces dependence on the individual $A_j$ and only logarithmic dependence on $B$. The nonvanishing argument and the arithmetic arithmetic height control are both necessary parts of this outline.

For practical equations the output is a finite, certified computation. Factor the equation in a number field, record the finitely many ideal and torsion possibilities, and express its factors in fundamental units. At a place where the original equation gives an exponentially small remainder, choose the principal logarithm and include a multiple of $2\pi i$ for the branch. Compare that upper bound with the effective lower bound to obtain an initial exponent bound. Question 5 shows this for a unit equation, and Question 6 shows why a rational argument of variable arithmetic height can still bound a common power exponent. Zero forms and degenerate families are separated before any estimate is applied.

A concrete two-logarithm example is $2^u-3^v=1$, with positive exponents. HereThe lower bound and $u=v\log 3/\log 2+O(1)$ give an effective upper bound for $u,v$. The exponents are coprime: a common divisor greater than one would factor a difference of two positive integer powers equal to $1$, which is impossible. For large $v$ the same upper estimate makes $u/v$ a continued-fraction convergent of $\log 3/\log 2$, since its error is smaller than $1/(2v^2)$. One checks only the finitely many convergents up to the proven bound, with small cases checked directly. Higher-dimensional problems use the LLL algorithm to obtain analogous improvements from the first large bound.

Every numerical logarithm must be evaluated with a certified error interval, so that a proposed reduction really is an inequality and an exact integer check can certify each surviving candidate. Continued fractions or lattice reduction accelerate the search; the prior effective upper bound is what makes the search complete. This explains how linear forms in logarithms of algebraic numbers turn qualitative finiteness into practical Diophantine algorithms, while also outlining the auxiliary machinery behind the estimates.