Author

Shaun M. Fallat

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Preprint Sep 2026

A proof of Borcea's quadratic variance conjecture

Let $p$ be a complex polynomial of degree $n\ge2$, and let $\sigma_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $\sigma_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, give...

Ludovick Bouthat, Shaun M. Fallat · 0 citations

Preprint Sep 2026

Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach

For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we p...

Jane Breen, Shaun M. Fallat, Johnna Parenteau · 0 citations