TY - JOUR

T1 - GALOIS GROUPS OF RANDOM ADDITIVE POLYNOMIALS

AU - Bary-Soroker, Lior

AU - Entin, Alexei

AU - McKemmie, Eilidh

N1 - Publisher Copyright:
© 2024 American Mathematical Society.

PY - 2024/3

Y1 - 2024/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85186915157&partnerID=8YFLogxK

U2 - 10.1090/tran/9098

DO - 10.1090/tran/9098

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AN - SCOPUS:85186915157

SN - 0002-9947

VL - 377

SP - 2231

EP - 2259

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -