Miscellanea II

It's time for more miscellanea! 

• Here are two very slick tricks for showing specific sums are positive. 
• A claw is another name for the complete bipartite graph $K_{1,3}$${\displaystyle K_{1,3}}$ (alternatively, a star graph comprising three edges, three leaves, and a central vertex). A graph is said to be claw-free if it has no induced subgraphs which are isomorphic to a claw. A huge amount of literature has been devoted to studying the structure and properties of claw-free graphs, which I'll leave the interested reader to look up on Wikipedia. Of note was the question of whether, for any claw-free graph $G$, the independence polynomial \[I_G(x):= \sum_{A \text{independent subset of} G} x^{|A|} \] has all roots real. There is a structure theory for such graphs, so one might reasonably think that some appeal to the structure theory is necessary. Not so, however! Chudnovsky--Seymour prove this using an ingenious and elementary method, which I highly recommend reading (it is also a short paper). This goes via characterising when a finite set of polynomials $p_i$, each of which has all roots real, has the property that. for any positive $c_i$. the polynomial \[\sum_i c_i p_i\] also has only real roots. It is especially remarkable that all the subclaims are fairly easy to prove, but finding the right sequence of these is where the insight lies. One could very well give the set of claims to a secondary school student and ask them to prove all of them and deduce the main result, which would be an interesting pedagogical test.
• Here's a fun problem about polynomials having only real roots.
• Bowditch gave a nice, conceptual, and short proof of the fact that a subquadratic isoperimetric inequality must be linear (this implies that a subquadratic Dehn function must be linear, i.e. define a hyperbolic group). I thank Shaked Bader, who extended the result to homological filling functions over any rings for one of her projects, for telling me about this.
• I learned about the Mason--Stothers theorem from a book of Serge Lang. Snyder's elementary proof is very nice and makes the result, in my opinion, a suitable olympiad problem (I have used it before). It implies Fermat's Last Theorem for function fields is true and much easier to prove (it seems to be a recurring theme that things are easier in the function field case). Lang's book also elaborates on the connections to the famous abc conjecture.

• Sam Fisher told me about the following: 
  Theorem: Let $R$ be a ring and let $M$ be a right $R$-module. Then $M$ is finitely presented if and only if the functor $M\otimes -$ preserves products: that is, for every family $(A_i)$ of left $R$-modules, the canonical map $M\otimes\prod A_i\to \prod M\otimes A_i$ is an isomorphism.
  
  I find this a very clean and appealing criterion, and it underlies his and Pablo Sanchez Peralta's proof of coherence of 2-dimensional RFRS groups with vanishing second $L^2$-Betti number. The proof can be found here.