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Explore the foundations of transcendence theory in this comprehensive lecture on auxiliary polynomials. Delve into the study of rationality and algebraicity properties of arithmetic and analytic quantities, starting with Hilbert's 7th problem and its solution by Gelfond and Schneider. Examine Baker's groundbreaking generalization of this result, focusing on the linear dependence of logarithms of algebraic numbers. Learn about the technique of auxiliary polynomials and its crucial role in proving these theorems. Investigate the structural rank conjecture and the significant contributions of Masser and Waldschmidt in providing lower bounds for the rank of logarithmic matrices. Gain insights into complex analysis and commutative algebra techniques used throughout the course. This 59-minute lecture, presented by Samit Dasgupta at the Hausdorff Center for Mathematics, serves as an excellent introduction to this fascinating area of mathematics.
