Mahler's method uses functional equations relating a function at and at to transport an auxiliary polynomial to points rapidly approaching zero. High vanishing order gives an upper bound which conflicts with an algebraic norm or height lower bound under assumed algebraicity of the values. Functional transcendence ensures that the auxiliary expression is not identically zero.
This lacunary series is analytic for and satisfies . At roots of unity of -power order, its radial tail diverges, giving a dense set of singularities on the unit circle and excluding algebraicity as a function. The Mahler method proves transcendence of its values at every algebraic with . At , reduced truncations have denominators and error less than twice their th power; for , the Roth theorem gives a shorter proof.
For , truncating the Fredholm series at gives a rational with denominator at most and positive error at most for large . The displayed strict inequality first rules out a rational value by the rational separation bound, and then rules out an algebraic irrational value by Roth theorem. Reduced denominators are at most and tend to infinity because the truncations are distinct. In particular and , , imply transcendence.
A nonzero integer polynomial of bidegree at most can cancel the first Taylor coefficients of the Fredholm series. Functional transcendence makes the resulting analytic function nonzero. If and were both algebraic, the relation would keep the auxiliary values in one fixed number field. Their Absolute logarithmic Weil height is at most , whereas their chosen absolute value is . The Liouville height inequality forces , contradicting its quadratic lower bound for sufficiently large .
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