The batch 20 to 22

Sendov.R_le_batch bounds every R n α for 20 ≤ n ≤ 22 by the elementary part and moment at n₀ = 20 together with the prefactor at n₁ = 22, so one moment and one certificate serve all 3 degrees. The certificate has degree 18, set by n₀ rather than n₁.

Feasibility at n₀ is proved rather than assumed: for n ≥ 36 it follows from 0 ≤ α ≤ 17, since A - c² increases with n. This matters because feasibility propagates upward in n, so it could not be inherited from the hypothesis at n.

The moment numerator Nmomc is checked against the packed recurrence (Sendov.pev_wsum_eq_of_packed), and the numerator Sendov.batchP 20 22 8 Lc Nmomc of 1 - bound is certified positive on [0, 21/2] by its Bernstein coefficients Bc (Sendov.pev_pos_of_bern). Every closed computation is evaluated by the kernel.

def Sendov.Batch20To22.Lc : ℤ

The common denominator L of the moment weights: j + 4 ∣ L for every j < 2k + 1.

Equations

• Sendov.Batch20To22.Lc = 232792560

def Sendov.Batch20To22.betac : ℤ

The base β at which polynomials in α are packed: it exceeds twice the absolute value of every coefficient of Nmomc and of the weighted row sum wsum Lc 0 (qrow …).

Equations

• Sendov.Batch20To22.betac = 4131209409209679854788987576321

def Sendov.Batch20To22.tauc : ℤ

The base τ at which the recurrence rows, evaluated at betac, are packed into a single integer exponentiation: it exceeds twice the absolute value of every row entry.

Equations

• One or more equations did not get rendered due to their size.

def Sendov.Batch20To22.Nmomc : List ℤ

The moment numerator at n₀ = 20, k = 8.

Equations

• One or more equations did not get rendered due to their size.

def Sendov.Batch20To22.Bc : List ℤ

Bernstein coefficients of 21 ^ 18 * batchP 20 22 8 Lc Nmomc on [0, 21/2].

Equations

• One or more equations did not get rendered due to their size.

theorem Sendov.Batch20To22.c_lo {α : ℝ} (hα : 0 ≤ α) : c 20 α = (114 + 13 * α - 2 * α ^ 2) / (38 * (3 + α))

theorem Sendov.Batch20To22.c_lo_nonneg {α : ℝ} (hα : 0 ≤ α) (hU : α ≤ 21 / 2) : 0 ≤ c 20 α

c is nonnegative at n₀ on the batch's α-range. This replaces feasibility at n₀, which for n₀ < 36 does not follow from α ≤ 17.

theorem Sendov.Batch20To22.pev_Nmomc (α : ℝ) : pev Nmomc α = pev (wsum Lc 0 (qrow (gg0 20) (gg1 20) (gg2 20) 8)) α

theorem Sendov.Batch20To22.integral_lo (α : ℝ) (hα : 0 ≤ α) : ∫ (t : ℝ) in 0..1, t ^ 3 * Q 20 α t ^ 8 = pev Nmomc α / (↑Lc * (2 * M 20 * (3 + α)) ^ 8)

theorem Sendov.Batch20To22.P_pos {α : ℝ} (hα : 0 ≤ α) (hU : 2 * α ≤ 21) : 0 < pev (batchP 20 22 8 Lc Nmomc) α

The certificate: batchP 20 22 8 Lc Nmomc is positive on [0, 21/2].

theorem Sendov.Batch20To22.finite_range {n : ℕ} (h0 : 20 ≤ n) (h1 : n ≤ 22) {α : ℝ} (hα : 0 ≤ α) (_hα' : α ≤ 17) (hfeas : c n α ^ 2 ≤ A n α) : R n α < 1

The batch 20 ≤ n ≤ 22.