[Paper Review] Explicit Polynomials Having the Higman-Sims Group as Galois Group over Q(t)

Dominik Barth, Andreas Wenz|arXiv (Cornell University)|Nov 14, 2016

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper constructs explicit degree-100 rational functions over ℚ(t) whose Galois groups are the Higman-Sims group HS and its automorphism group Aut(HS), using a Belyi map of genus 0 with monodromy corresponding to a rigid rational triple. The authors compute explicit polynomials via algebraic and geometric methods, verifying the monodromy group through permutation group theory and discriminant analysis, thus solving the inverse Galois problem for HS over ℚ(t).

ABSTRACT

We compute explicit polynomials having the sporadic Higman-Sims group HS and its automorphism group Aut(HS) as Galois groups over the rational function field Q(t).

Motivation & Objective

• To solve the inverse Galois problem for the sporadic Higman-Sims group HS and its automorphism group Aut(HS) over the rational function field ℚ(t).
• To construct explicit rational functions f(X) = p(X)/q(X) of degree 100 with monodromy group isomorphic to Aut(HS).
• To verify that the Galois group of the polynomial p(X) - tq(X) ∈ ℚ(t)[X] is precisely Aut(HS) using algebraic and geometric techniques.
• To provide two distinct Belyi maps of degree 100 corresponding to the two rigid rational generating triples of Aut(HS) of genus 0.
• To demonstrate a method for computing higher-degree Belyi maps using permutation triples and rational rigidity, applicable to sporadic groups.

Proposed method

• Construct a Belyi map f: ℙ¹(ℂ) → ℙ¹(ℂ) of degree 100 with monodromy group isomorphic to Aut(HS), based on a rigid rational triple (x, y, z) in S₁₀₀ satisfying x·y·z = 1 and genus 0.
• Define f(X) = p(X)/q(X) = 1 + r(X)/q(X), where p(X), q(X), and r(X) are explicitly computed polynomials with integer coefficients and specific factorizations.
• Use Magma to verify that the permutation triple (x, y, z) generates Aut(HS), satisfies the braid relation, and has genus 0, ensuring the existence of a unique Belyi map over ℚ.
• Apply the Riemann-Hurwitz formula to confirm that f is a three-point branched cover ramified over 0, 1, and ∞, with cycle types matching the given permutations.
• Verify monodromy algebraically by analyzing the Galois group of p(X) - tq(X) ∈ ℚ(t)[X], showing it is a primitive rank-3 permutation group with subdegrees 1, 22, and 77.
• Determine whether the Galois group is HS or Aut(HS) by checking whether the discriminant of p(X) - tq(X) is a square in ℚ(t); it is not, confirming the group is Aut(HS).

Experimental results

Research questions

• RQ1 Can an explicit Belyi map of degree 100 with monodromy group isomorphic to Aut(HS) be constructed over ℚ(t)?
• RQ2 Does the Galois group of the polynomial p(X) - tq(X) ∈ ℚ(t)[X] equal Aut(HS), and how can this be verified algebraically?
• RQ3 Are there two distinct rigid rational generating triples for Aut(HS) of genus 0, and do they yield different Belyi maps with the same monodromy group?
• RQ4 Can the discriminant of the polynomial p(X) - tq(X) be used to distinguish between the Galois groups HS and Aut(HS) over ℚ(t)?
• RQ5 What is the structure of the dessin d’enfant associated with the Belyi map, and how does it reflect the monodromy group?

Key findings

• The authors construct an explicit Belyi map f(X) = p(X)/q(X) of degree 100 over ℚ(t), with monodromy group isomorphic to Aut(HS), derived from a rigid rational triple of genus 0.
• The polynomial p(X) - tq(X) ∈ ℚ(t)[X] has Galois group isomorphic to Aut(HS), confirmed by discriminant analysis showing the discriminant is not a square in ℚ(t).
• A second Belyi map of degree 100 is constructed from a second rigid rational triple, yielding another explicit polynomial with Galois group Aut(HS) over ℚ(t).
• The Galois group of p(X) - (2t² + 1)q(X) is shown to be HS, as the discriminant is a square in ℚ(t), distinguishing it from the previous case.
• The monodromy group is verified both geometrically (via the dessin d’enfant and cycle structure of x and y) and algebraically (via permutation group theory and subdegree analysis).
• The rational function f(X) = p(X)/q(X) is confirmed to be a three-point branched cover of ℙ¹(ℂ) ramified over 0, 1, and ∞, with ramification types matching the cycle structures of x, y, and z.