September 26, Wednesday, 18:30, room 306
shinymath, · Categories: Без рубрикиAlexander Esterov
Solvability of equations by radicals and classification of lattice polytopes
The celebrated Abel–Ruffini theorem states that it is impossible to express the solutions of the general polynomial equation of degree d>4 in terms of its coefficients, using the arithmetic operations and arithmetic roots.
We shall present the proof of this theorem based on the technique of topological Galois theory (different from the well known “Arnold’s proof”). We shall also generalize this theorem to systems of polynomial equations: we shall see that the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of volume at most 4, which is essentially finite (i.e. we can explicitly list the 34 “elementary” lattice polytopes of volume 4 and the corresponding 34 “elementary” solvable systems of equations, from which all the others can be constructed).
The aim of the talk is (1) to demonstrate one instance of the relation between algebraic and convex geometry based on the notion of Newton polytopes and (2) to provide the background for a subsequent talk about the Galois theory of Schubert calculus.
No previous knowledge of Galois theory is required.