Logarithmic form

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Any formula written in terms of logarithms may be said to be in logarithmic form.

[edit] Number theory

In number theory a logarithmic form or linear form in logarithms is assumed to be a finite sum

Σ α_i log β_i = Λ

where the α_i and β_i are algebraic numbers. In case of β_i a complex number, one has to allow log to denote some definite branch of the logarithm function in the complex plane. The basic problem attacked in Alan Baker's work is to supply lower bounds for |Λ|, in cases where Λ ≠ 0. This is in terms of quantities A and B, respectively bounding the heights of the α_i and β_i. This work supplied many results on diophantine equations, amongst other applications. It has been suitably generalised to abelian varieties.

[edit] Logarithmic differential forms

In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a 1-form that, locally at least, can be written

$\frac{df}{f}$

for some meromorphic function (resp. rational function) f. That is, for some open covering, there are local representations of this differential form as a logarithmic derivative (modified slightly with the exterior derivative d in place of the usual differential operator D). These forms are quite highly constrained in their behaviour. For example on a Riemann surface it follows that they have simple poles, and everywhere integer residues at them. In higher dimension one needs the Poincaré residue to formulate their distinctive behaviour at places where f takes the value 0 or ∞.

Classically, for example in elliptic function theory, the logarithmic differential forms were recognised as complementary to the differentials of the first kind. They were sometimes called differentials of the second kind (and, with an unfortunate inconsistency, also sometimes of the third kind). The classical theory has now been subsumed as an aspect of Hodge theory. For a Riemann surface S, for example, the differentials of the first kind account for the term H^0,1 in H¹(S), when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group H⁰(S,Ω); this is tautologous considering their definition. The H^1,0 direct summand in H¹(S), as well as being interpreted as H¹(S,O) where O is the sheaf of holomorphic functions on S, can be identified more concretely with a vector space of logarithmic differentials.