Samit Dasgupta: An introduction to to auxiliary polynomials in transcendence theory, Lecture III

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Broadly speaking, transcendence theory is the study of the rationality or algebraicity properties of quantities of arithmetic or analytic interest. For example, Hilbert’s 7th problem asked ”Is a b always transcendental if a 6= 0, 1 is algebraic and b is irrational algebraic?” An affirmative answer was proven by Gelfond and Schneider in 1934-35. In the late 1960s, Baker generalized this result in spectacular fashion, proving that if a set of logarithms of algebraic numbers is linearly dependent over the algebraic numbers, then it is in fact linearly dependent over the rational numbers.
All of these results used the technique of auxiliary polynomials. To be precise, Baker showed that if a set of logarithms of algebraic numbers is linearly dependent over the algebraic numbers, then there exists a polynomial whose vanishing set is of a particular form; the existence of this polynomial is then used to deduce the linear dependence of the original logarithms over the rational numbers.

We will begin the course by proving Baker’s Theorem. Next, we will describe the structural rank conjecture, which predicts the rank of a matrix of logarithms of algebraic numbers. We will prove the beautiful theorem of Masser and Waldschmidt that provides a lower bound on this rank, giving
a partial result toward the structural rank conjecture. Waldschmidt’s theorem gives the construction of a certain auxiliary polynomial, and Masser’s theorem provides the desired lower bound given the existence of this auxiliary polynomial. Techniques used in the course will include complex analysis and commutative algebra.
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