Speaker
Description
Polynomial factorization is classically studied within commutative polynomial rings, where irreducibility is an intrinsic algebraic property. In this talk, we present a linear-algebraic approach to factorization via entangled polynomial rings, in which polynomials are represented by structured matrices and analyzed using tools from matrix theory.
By embedding a polynomial into a family of $m \times m$ matrices (through $m$-nomials and their finite representations as Szab\'o rings), questions of reducibility are translated into questions about matrix factorizations. In this setting, reducibility is detected using the $m$-terminant, defined as the determinant of the associated matrix. Circulant matrices and their eigenvalues play a central role in determining when such factorizations occur.
We focus on the polynomials
$$
f_p(x) = 1 + x + x^2 + \cdots + x^{p-1},
$$
which are irreducible in $\mathbb{Q}[x]$ for prime $p$. Using determinant and eigenvalue computations, we show that while these polynomials remain irreducible for all $m < p$, their matrix representations become reducible at $m = p$. This leads to the notion of \emph{valence}, the minimal matrix size at which reducibility occurs, and establishes that the valence of $f_p(x)$ is exactly $p$ for all primes $p \ge 3$.
This perspective reframes polynomial factorization as a problem in linear algebra over rings, illustrating how matrix representations, determinants, and eigenvalues uncover hidden algebraic structure beyond the classical setting.