The Geometry of Hilbert's 13th problem

Special Seminar on Hilbert's 13th Problem I
Topic:The Geometry of Hilbert's 13th problem
Speaker:Jesse Wolfson
Affiliation:University of California, Irvine
Date:Thursday, December 5
Time/Room:1:00pm - 1:55pm/Simonyi Hall 101
Video Link:https://video.ias.edu/special/2019/1205-JesseWolfson

Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables respectively (such formulas were first constructed by Hamilton). In a little-known paper, Hilbert sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9 polynomial. In this talk I’ll recall Klein and Hilbert's geometric reformulation of solving polynomials, explain the gaps in Hilbert's sketch and how we can fill these using modern methods. As a result, we obtain best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial for all n, improving results of Segre and Brauer.