Uniform irreducibility and Galois groups of generalized Laguerre polynomials
Source: hexagonmath.org/abs/2611.00087v1
We prove that the generalized Laguerre polynomial $H_{n,s}$ is irreducible over $\mathbb Q$ and has Galois group containing $A_n$ for every positive integer $s$, once $n$ exceeds an effective absolute constant. Equivalently, every numerator and denominator in the exponential Pad\'e table has these properties once its own degree is sufficiently large. The exceptional positive integral pairs form an effectively computable finite set. The proof combines integral traces at degenerate fibers, Pad\'e resultants, and an energy inequality for Galois-invariant positive semidefinite forms on the roots.
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