Oppermann intervals in F_q[t]

Galois obstructions, exact counts for small degree, and the Oppermann subgroup in characteristic 2

Ruqing Chen. DOI: 10.5281/zenodo.23353949

Paper: paper/oppermann_function_fields.tex / .pdf (23 pages).

For monic f of degree d, the Legendre interval {f² + s : deg s ≤ d} splits into q Oppermann sub-intervals

O_{f,c} = {f² + c·f + s : deg s < d}, c ∈ F_q,

each of size q^d. For q = 2 these are the two halves, the analogue of (n², n²+n) and (n²+n, (n+1)²). Does every O_{f,c} contain an irreducible polynomial?

Main results

Characteristic 2: no. The failures are explained by three Galois obstructions (§3).

• (A) d = 1: the c = 0 piece consists of squares.
• (B) d = 2: Frobenius acts on the roots of an additive polynomial through AGLâ‚‚(Fâ‚‚) ≅ Sâ‚„. The exact counts are q²/4, 0, q(q−2)/4 and q(q−1)/2. Empty sub-intervals occur exactly for q = 2 and q = 4^j.
• (C) q = 2, d = 4: the Stickelberger–Swan sign of Frobenius obstructs.
• Dead zone (§3.4): for r ≥ 3, AGL_r(Fâ‚‚) contains no 2^r-cycle. Hence no affine polynomial of degree 2^r ≥ 8 is irreducible, and the mechanism of (B) does not extend to d ≥ 4.

Exact counts for small degree

• Odd q, d = 2 (§4): a ternary trace form on F_{qâ�´} gives Ï€ = (q² − 1 − (q − η(−1))η(2e))/4 > 0.
• Characteristic 2, d = 3 (§5): a count in the field tower F_q ⊂ F_{q³} ⊂ F_{qâ�¶}, together with a cubic Gauss sum, gives every class. Let c be the t³-coefficient and ε = (−1)^{k/2}.
  • k odd: Ï€ = q(q² − 1)/6 for every class.
  • k even, c = 0: Ï€ = q(q − 1)²/6.
  • k even, c a cube: Ï€ = q(q² + 1 − 2ε√q)/6.
  • k even, c a non-cube: Ï€ = q(q² + 1 + ε√q)/6.

Large q and computation (§6)

• Large q: for fixed d ≥ 3, Bank–Bary-Soroker–Rosenzweig together with the d = 3 count gives nonemptiness once q is large.
• Odd q: exact Hayes-FFT counts find every sub-interval nonempty for q = 3, d ≤ 16; q = 5, d ≤ 10; q = 7, d ≤ 9; q = 9, d ≤ 8; q = 11, d ≤ 7; q = 13, d ≤ 6.
• Characteristic 2: the same computation finds no empty sub-intervals beyond (A)–(C) for q = 2, d ≤ 26; q = 4, d ≤ 13; q = 8, d ≤ 7; q = 16, d ≤ 5.
• Weil bound: it sits at the square-root barrier. At d = 2 every level-2 L-function in characteristic 2 is rigid, with γâ�´ = (−1)^k q².

The Oppermann subgroup in characteristic 2 (§2.1, §7)

• Structure: H_d = ker(P ↦ Sâ‚�, S₃, …), so G_d/H_d ≅ F_q^{⌊d/2⌋}. The annihilator's explicit-formula coefficients are Artin–Schreier sums.
• Mean drift: E_H[Ψ] − q^d is 0 for d = 2, 3 and for d = 4 with k even. It is −2q² + 2q for d = 4 with k odd, and q² − q for d = 5.
• Variance: it is a coherent sum over annihilator cosets.
  • d = 2: Var_H/q² = q − 1, against Var_G/q² = 1 − 1/q.
  • d = 3: Var_H/q³ = 1 − 1/q (k odd) or 3(1 − 1/q) (k even).
  • For d ≥ 4 there is a numerical census, compared with the Keating–Rudnick value d − 1.

Files

script (code/)	output (data/, figures/, paper/)	paper
hayes_oppermann.py	hayes_oppermann_census.txt	Thms 1.2, 1.3
verify_pointwise_oppermann.py, compare_bruteforce.py	oppermann_census.txt, compare_bruteforce.txt	cross-checks
char2_d2_blocks.py, char2_d2_theorem.py	char2_d2_blocks.txt, char2_d2_theorem.txt	Thm 1.1(B)
swan_parity.py	swan_parity.txt	Thm 1.1(C), Table 1
verify_additive_slices.py	additive_slices.txt	Thm 3.6, Cor 3.7
odd_d2_formula.py	odd_d2_formula.txt	Thm 1.4
char2_d3_p3zero.py, char2_d3_all.py	char2_d3_p3zero.txt, char2_d3_all.txt	Thm 5.2
cubic_gauss_m.py, make_d3_table.py	cubic_gauss_m.txt, paper/tab_d3.tex	Thm 5.3, Cor 5.4, Table 2
char2_d2_rigidity.py	char2_d2_rigidity.txt	Prop 6.1
verify_level3_cosets.py	level3_cosets.txt	Prop 7.6 (level-3 coset sums; γ⁶ = q³ on the non-cube cosets)
subgroup_lib.py, verify_subgroup_mean.py	verify_subgroup_mean.txt	Thms 2.4, 2.5, 7.3
subgroup_variance_census.py, verify_variance23.py	subgroup_variance.{txt,json}, verify_variance23.txt	Thm 7.5
make_figure.py, make_figure2.py, make_tables.py	figures/oppermann_min_ratio.pdf, figures/oppermann_char2_rigidity.pdf, paper/census_tables.tex	Figs 1, 2
make_subgroup_figures.py, make_subgroup_tables.py	figures/subgroup_mean_gauss.pdf, figures/subgroup_variance.pdf, paper/tab_subgroup_*.tex	Figs 3, 4
hayes_fft_fq.py, fqlib.py, small_q_gf.py	—	from Part I. The only change is extra moduli in fqlib.py for q = 32, …, 256

notes/REPORT.md contains the working notes of the first round.

Run with:

cd code
python hayes_oppermann.py 3 1 16 <tmpdir>      # q d_min d_max [memmap dir]
python verify_subgroup_mean.py > ../data/verify_subgroup_mean.txt
python subgroup_variance_census.py <tmpdir>
python cubic_gauss_m.py 14 > ../data/cubic_gauss_m.txt
python make_figure.py; python make_figure2.py; python make_subgroup_figures.py
python make_tables.py; python make_d3_table.py; python make_subgroup_tables.py
cd ../paper && pdflatex oppermann_function_fields.tex