GALOIS GROUPS OF RANDOM ADDITIVE POLYNOMIALS

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Abstract

We study the distribution of the Galois group of a random qadditive polynomial over a rational function field: For q a power of a prime p, let f = Xqn +an−1Xqn−1 +. . .+a1Xq +a0X be a random polynomial chosen uniformly from the set of q-additive polynomials of degree n and height d, that is, the coefficients are independent uniform polynomials of degree deg ai ≤ d. The Galois group Gf is a random subgroup of GLn(q). Our main result shows that Gf is almost surely large as d, q are fixed and n → ∞. For example, we give necessary and sufficient conditions so that SLn(q) ≤ Gf asymptotically almost surely. Our proof uses the classification of maximal subgroups of GLn(q). We also consider the limits: q, n fixed, d → ∞ and d, n fixed, q → ∞, which are more elementary.

Original languageEnglish
Pages (from-to)2231-2259
Number of pages29
JournalTransactions of the American Mathematical Society
Volume377
Issue number3
DOIs
StatePublished - Mar 2024

Funding

FundersFunder number
Israel Science Foundation702/19, 2507/19

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