Table 3 Checking conditions (18)\(_{r,s}\) for Example 5
From: On algorithms testing positivity of real symmetric polynomials
Restrictions | Input n:= | Input (r,s):= | (18)\(_{r,s}\) holds since |
|---|---|---|---|
r,s ≥ 6 | r + s | (p + 6,q + 6) | α,β,γ ≥ 0 |
s ≥ 3, r ≥ 3s − 2 | r + s | (3s − 2 + p,q + 3) | P,Q,α,β ≥ 0 |
3 ≤ s ≤ 5, s + 1 ≤ r ≤ 3s − 3 | r + s | \((\frac{p(s+1)+3s-3}{p+1},\frac{3q+5}{q+1} )\) | P,R ≥ 0 |
r ≥ 3, s ≥ r + 1 | r + s | (p + 3,r + 1 + q) | β,γ,P ≥ 0 ≥ Q |
r = s ≥ 3 | r + s | (p + 3,r) | P,R ≥ 0 |
r = 2, s ≥ 3 | r + s | (2,q + 3) | α = 0 |
s = 2, r ≥ 3 | r + s | (p + 3,2) | P ≥ 0 = R |
s = 1, r ≥ 5 | r + s + q | (p + 5,1) | P,R ≥ 0 |
s = 1, r = 4 | r + s + q | (4,1) | P,R ≥ 0 |
r = 1, s ≥ 6 | r + s + p | (1,q + 6) | P,R ≥ 0 |
r = 1, s = 4 | r + s + p | (1,4) | P,R ≥ 0 |
r = 1, s = 5, n ≥ 7 | p + 7 | (1,5) | P,R ≥ 0 |
r = 1, s = 5, n = 6 | 6 | (1,5) | Δ<0 |