Legendre's Conjecture in Function Fields: Full Monodromy, Twisted Root Varieties, and Explicit Bounds

For a monic polynomial f in F_q[t] of degree d, the Legendre interval I_f = {f^2 + s : deg s <= d} is the function field analogue of [n^2, (n+1)^2]. This revised version (v6) corrects and extends v5 (doi:10.5281/zenodo.18705744). The asymptotic count N_irr = q^{d+1}/(2d) + O(q^{d+1/2}) of irreducible polynomials in I_f is a special case of the short-interval theorem of Bank, Bary-Soroker and Rosenzweig (Duke Math. J. 2015). Version v5 presented it as new; v6 states this plainly. New in v6:(A) A self-contained proof that the geometric monodromy group is the full symmetric group S_2d when char F_q > 2d, without the squarefree hypothesis on f used in v5.(B) An exact identity 2d N_irr = #W(F_q) - E, where W is an explicit twisted root variety of dimension d+1 and degree at most (d-1)!, and 0 <= E < q^{d+1}/(q-1).(C) Explicit, fully proved thresholds q_0(d) beyond which every Legendre interval contains an irreducible polynomial, via the Cafure-Matera explicit Lang-Weil bound (e.g. q_0(4) ≈ 1.65·10^4) and, separately, via Katz's Betti-number bound and Deligne's theorem.(D) Closed formulas for d <= 3 valid for every admissible q; the d = 3 case agrees with Kuz'min's formula for two prescribed coefficients. Computations for d <= 6 (186 exact counts, verification of the identity by enumerating F_{q^{2d}}, and an exhaustive search over all f for small q) support every statement. An appendix lists the errors in v5 and how each is fixed. The case of fixed q and d -> infinity remains open. Code, data, figures and a referee report on v5: https://github.com/Ruqing1963/legendre-function-field