Email Gateway (Aug 23 2022 at 08:51): Email Gateway (Aug 23 2022 at 08:51):From: "Thiemann, René" <Rene.Thiemann@uibk.ac.at>
Dear all,
there is second new entry by Manuel Eberl:
Power Sum Polynomials
This article provides a formalisation of the symmetric multivariate polynomials known
as power sum polynomials. These are of the form p_n(X_1,...,X_k) = X_1^n + … + X_k^n.
A formal proof of the Girard–Newton Theorem is also given. This theorem relates the power sum polynomials to the elementary symmetric polynomials s_k in the form of a recurrence
relation (-1)^k * k * s_k = Sum_{i in [0,k)} (-1)^i * s_i * p_{k-i}.
As an application, this is then used to solve a generalised form of a puzzle given as
an exercise in Dummit and Foote's Abstract Algebra: For k complex unknowns x_1, ..., x_k,
define p_j := x_1^j + … + x_k^j. Then for each vector a in ℂ^k, show that there is exactly one
solution to the system p_1 = a_1, …, p_k = a_k up to permutation of the x_i and determine the
value of p_i for i>k.
Enjoy even further,
René
PS: Here are the links to both of Manuel’s entries:
https://www.isa-afp.org/entries/Power_Sum_Polynomials.html
https://www.isa-afp.org/entries/Gaussian_Integers.html
Last updated: Nov 21 2024 at 12:39 UTC