Probabilistic Galois theory in function fields (Q6597210)

Probabilistic Galois theory examines the statistical properties of the Galois group of random polynomials over global fields. In the classical setting with integer coefficients over \(\mathbb{Q}\), random polynomials are almost surely irreducible, with their Galois group isomorphic to the symmetric group \(S_n\). This result holds in the large box model, where coefficients are drawn from a fixed set and the degree is fixed as the set size increases. In contrast, the small box model considers polynomials with coefficients from a more limited set.\N\NThis work explores probabilistic Galois theory over the rational function field \(F_q(x)\), where \(q\) is a prime power. The authors investigate the irreducibility and Galois group of random polynomials, focusing on those of the form \( f(x,y) = y^n + a_{n-1}(x)y^{n-1} + \cdots + a_0(x) \), where the \(a_i(x)\) are polynomials in \(\mathbb{F}_q[x]\) of degree at most \(d\), chosen independently at random. In the large box model, they show that for fixed degree \(n\) and prime power \(q\), \(f(x,y)\) is almost surely irreducible and separable over \(\mathbb{F}_q(x)\), with the Galois group almost surely isomorphic to \(S_n\) as \(d \to \infty\), paralleling Hilbert's Irreducibility Theorem for function fields.\N\NIn the small box model, where \(d\) is fixed and \(n\) grows large, the probability of irreducibility approaches \(1 - \frac{1}{q^d}\) as \(n \to \infty\). This extends Carlitz's results with a novel approach. The authors also analyze the distribution of the Galois group for irreducible polynomials. Theorem 1.4 establishes that, as \(n \to \infty\), the Galois group of \(f(x,y)\) is almost surely isomorphic to either \(S_n\) or \(A_n\), though distinguishing between these remains a complex problem.\N\NThe authors propose a uniform version of the polynomial Chowla Conjecture. Assuming this conjecture, Theorem 1.6 strengthens Theorem 1.4 by showing that, for an irreducible \(f(x,y)\) over \(\mathbb{F}_q(x)\), its Galois group is almost surely \(S_n\) when \(q\) is an odd prime power, as \(n \to \infty\).\N\NIn Theorem 1.8, the authors extend their analysis to reducible polynomials, proving that the Galois group of the splitting field of \(f(x,y)\) over \(\mathbb{F}_q(x)\) is almost surely a Galois extension of \(\mathbb{F}_q(x)\), as \(n \to \infty\). Additional strengthened results follow from assuming their conjecture.\N\NThis paper presents significant results across both large and small box models, with a focus on Galois group distributions and an extension of the analysis to reducible polynomials.