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1. Almost All Alternating Groups are Invariably Generated by Two Elements of Prime Order

   https://doi.org/10.1093/imrn/rnac354  

   Teräväinen, Joni (January 2023, International Mathematics Research Notices)

   Abstract We show that for all $$n\leq X$$ apart from $$O(X\exp (-c(\log X)^{1/2}(\log \log X)^{1/2}))$$ exceptions, the alternating group $$A_{n}$$ is invariably generated by two elements of prime order. This answers (in a quantitative form) a question of Guralnick, Shareshian, and Woodroofe. 

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2. A STABILITY VERSION OF THE GAUSS–LUCAS THEOREM AND APPLICATIONS

   https://doi.org/10.1017/S1446788719000284  

   STEINERBERGER, STEFAN (September 2019, Journal of the Australian Mathematical Society)

   Let $$p:\mathbb{C}\rightarrow \mathbb{C}$$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $$p^{\prime }(z)=0$$ , are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $$p$$ has $n+m$ roots, where $$n$$ are inside the unit disk, $$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$ then $$p^{\prime }$$ has $n-1$ roots inside the unit disk and $$m$$ roots at distance at least $(dn-m)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for $$n$$ sufficiently large, each of the $$m$$ roots has a critical point at distance $${\sim}n^{-1}$$ . 

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3. Additive Energies on Discrete Cubes

   de Dios Pont, Jaume; Greenfeld, Rachel; Ivanisvili, Paata; Madrid, Jose (September 2023, Discrete analysis)

   Gowers, Tim (Ed.)

   We prove that for $$d\geq 0$$ and $$k\geq 2$$, for any subset $$A$$ of a discrete cube $$\{0,1\}^d$$, the $k-$higher energy of $$A$$ (i.e., the number of $2k-$tuples $$(a_1,a_2,\dots,a_{2k})$$ in $$A^{2k}$$ with $$a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$$) is at most $$|A|^{\log_{2}(2^k+2)}$$, and $$\log_{2}(2^k+2)$$ is the best possible exponent. We also show that if $$d\geq 0$$ and $$2\leq k\leq 10$$, for any subset $$A$$ of a discrete cube $$\{0,1\}^d$$, the $k-$additive energy of $$A$$ (i.e., the number of $2k-$tuples $$(a_1,a_2,\dots,a_{2k})$$ in $$A^{2k}$$ with $$a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$$) is at most $$|A|^{\log_2{ \binom{2k}{k}}}$$, and $$\log_2{ \binom{2k}{k}}$$ is the best possible exponent. We discuss the analogous problems for the sets $$\{0,1,\dots,n\}^d$$ for $$n\geq2$$. 

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4. A Generalization of the Bollobás Set Pairs Inequality

   https://doi.org/10.37236/9627  

   O'Neill, Jason; Verstraete, Jacques (July 2021, The Electronic Journal of Combinatorics)

   The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $$n \geqslant k \geqslant t \geqslant 2$$, we consider a collection of $$k$$ families $$\mathcal{A}_i: 1 \leq i \leqslant k$$ where $$\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$$ so that $$A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing$$ if and only if there are at least $$t$$ distinct indices $$i_1,i_2,\dots,i_k$$. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size $$\beta_{k,t}(n)$$ of the families with ground set $[n]$. 

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5. Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

   https://doi.org/10.1093/imrn/rnz261  

   Billey, Sara C.; Rhoades, Brendon; Tewari, Vasu (November 2019, International Mathematics Research Notices)

   Abstract Let $$k \leq n$$ be positive integers, and let $$X_n = (x_1, \dots , x_n)$$ be a list of $$n$$ variables. The Boolean product polynomial$$B_{n,k}(X_n)$$ is the product of the linear forms $$\sum _{i \in S} x_i$$, where $$S$$ ranges over all $$k$$-element subsets of $$\{1, 2, \dots , n\}$$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $$B_{n,k}(X_n)$$ for certain $$k$$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $$B_{n,n-1}(X_n)$$ to a bigraded action of the symmetric group $${\mathfrak{S}}_n$$ on a divergence free quotient of superspace. 

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