Perlen-Kolloquium: Arnaud Beauville (University of Nice)

How many parameters are needed to define the general equation of degree n?

The equation xⁿ +a₁xⁿ⁻¹ +...+a_n = 0 depends on n coefficients a₁,...,a_n, but we all know that after a change of variable x → x+a we can assume a₁ = 0, so only n−1 parameters are needed. Can we get further? After some work of Hermite and others the subject lay dormant till 1997, when Buhler and Reichstein introduced the notion of essential dimension of a group. I will explain what it is, and how results of algebraic geometry (some of them quite involved) allow to solve the problem in degree ≤ 7. 

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