Newton’s sums, necklace congruences, and zeta functions II

In a previous post I gave essentially the following definition: given a discrete dynamical system, i.e. a space and a function , and under the assumption that has a finite number of fixed points for all , we define the dynamical zeta function to be the formal power series . What I didn’t do was [...]

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Newton’s sums, necklace congruences, and zeta functions

The goal of this post is to give a purely combinatorial proof of Newton’s sums which would have interrupted the flow of the previous post. Recall that, in the notation of the previous post, Newton’s sums (also known as the first Newton-Girard identity) state that . One way to motivate a combinatorial proof is to [...]

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Dynkin diagrams and the Mahler measure problem

Funnily enough, a few days after I wrote the previous post, I was linked to a graph theory paper where one of the first results cited, which was clearly well-known to the authors, is the following remarkable generalization of what I tried to do: Theorem: The only connected simple graphs with spectral radius less than [...]

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