When nature hands you an algebraic function, you have two tasks: 1) find a simple solution and 2) prove no simpler solution is possible. In this example driven talk, I’ll focus on 1), reviewing classical work of Hamilton, Klein, Hilbert and Brauer which used algebra, invariant theory, algebraic geometry and uniformization to produce explicit solutions, which give current upper bounds on the minimal number of variables needed to solve the general degree n polynomial or classical examples such as the equation of the 27 lines on a cubic surface. Time permitting, I’ll describe joint work with Benson Farb extending Hilbert’s idea of using lines on a cubic surface to solve polynomials of degree 9 and above.
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