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By Patrice Philippon, Michel Waldschmidt. Chapter 18 of New Advances in Transcendence Theory, pages 280–312, Cambridge University Press.

Author's primary text: webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Durham1986PW.pdf

Theorem 1.1, printed pages 280–281, treats algebraic coefficients including the constant term. Theorems 2.1–2.2, printed pages 282–283, distinguish general algebraic coefficients from the homogeneous rational-integer case and give the refined coefficient parameter.

Michel Waldschmidt works on transcendental number theory and Diophantine approximation. With Patrice Philippon, he wrote quantitative results for linear forms in logarithms of algebraic numbers of algebraic numbers.^[ref]

A nonzero expression such as $b_1\log {\alpha }_1+b_2\log {\alpha }_2$ can be small, but lower bounds limit how small it can be in terms of the algebraic data and coefficient sizes. Such bounds convert qualitative nonvanishing into numerical restrictions useful in solving Diophantine equations. The algebraic-coefficient and integer-coefficient cases require distinct estimates.^[ref]

Patrice Philippon works on transcendental number theory. His joint chapter with Michel Waldschmidt gives lower bounds for linear forms in logarithms of algebraic numbers, explicitly distinguishing general algebraic coefficients from rational-integer coefficients.^[ref]

The central arithmetic question is quantitative: once a logarithmic combination is known to be nonzero, how closely can it approach zero? An explicit bound links Diophantine approximation to finite calculations and can help restrict candidate solutions of Diophantine equations.