Linear form in logarithms of algebraic numbers

Baker's theorem gives transcendence and quantitative nonvanishing results for such logarithmic forms.

A homogeneous linear form in logarithms of nonzero algebraic numbers has the form

If chosen logarithms of algebraic numbers are linearly independent over $\mathbb{Q}$, then they, together with one, are linearly independent over $\overline{\mathbb{Q}}$. More generally a nonzero homogeneous linear combination with algebraic coefficients is transcendental. Quantitative lower bounds for integer coefficients are a separate, effective aspect of logarithmic-form theory.

Choose logarithms of nonzero algebraic numbers ${\alpha }_1,\dots ,{\alpha }_n$ and algebraic numbers ${\beta }_0,\dots ,{\beta }_n$, and putLet $A_j$ bound $10$, the naive polynomial height of the minimal polynomial of ${\alpha }_j$, and $\exp (|\log {\alpha }_j|)$. Let $B$ bound all the corresponding heights of the ${\beta }_j$ and all $\log A_j$. There is an effective constant $C$, depending only on $n$ and the degree of the number field generated by the data, such that

This quantitative refinement belongs to the theory of Baker's theorem; its dependence on the scaled coefficient bound is more specific than that broad theorem.

Choose logarithms of nonzero algebraic numbers ${\alpha }_1,\dots ,{\alpha }_n$, putand choose $A_i\ge 10$ large enough to bound the naive polynomial height of the minimal polynomial of ${\alpha }_i$ and $\exp (|\log {\alpha }_i|)$. After relabelling the terms if useful, setThere is an effectively computable constant $C$, depending only on $n$ and the degree of the number field generated by the ${\alpha }_i$, such that

The division by $\log A_n$ is the useful refinement over the general lower bound for a linear form in logarithms when one algebraic number has a large height correlated with a coefficient.

For fixed positive $a,b$ and fixed nonzero $c$, with $x\ne y$, put $Y=\max (x,y)$ and take $\Lambda =\log (a/b)+n\log (x/y)$, swapping the variables and coefficients if necessary. The equation gives $\log |\Lambda |\le -n\log Y+O(1)$ when the larger-variable power is sufficiently large. A two-logarithm lower bound gives $\log |\Lambda |\ge -C(a,b)\log Y\log (2n)$. Hence $n\le C\log (2n)+O(1)$, so $n$ is bounded. The cases $x=y$ and a bounded larger-variable power are elementary. For each remaining exponent, the Thue theorem for a binomial form supplies finitely many pairs. The constant-power exception $x=y=1$, $a-b=c$, must be removed.

Let $L$ be a fixed number field and let $V\subset L$ be a fixed rational linear subspace for which the associated norm form has no unit-family degeneracy. There are effective constants $A,C>0$ such that every $z\in V\cap {\mathcal{O}}_L$ with nonzero field norm $m$ satisfiesTo prove this, factor the principal ideal $(z)$, write its generators as a bounded factor times powers of fundamental units, use the linear relations defining $V$, and apply the Baker lower bound for a homogeneous linear form in logarithms to bound the unit exponents by $O(\log |m|)$. For $L=\mathbb{Q}(\alpha )K$ with $[\mathbb{Q}(\alpha ):\mathbb{Q}]=3$, $[K:\mathbb{Q}]=2$, and $V={\mathrm{span}}_{\mathbb{Q}}\{1,\alpha ,\omega \}$ for a basis $1,\omega$ of $K$, the coprime degrees rule out the degeneracy.