Untitled

A proof of Wright's conjecture

“…Although we use radii polynomials as well, our approach differs significantly from the computer-assisted setting mentioned above. While we do engage a computer (namely the Mathematica file [21]) to optimize our quantitative results, the analysis is performed essentially in terms of pencil-and-paper mathematics (in particular, our operators do not involve any floating point numbers). In our current setup we employ three radii as a priori unknown variables, which builds on an idea introduced in [20].…”

“…We check the hypothesis is indeed satisfied by using interval arithmetic. All details are provided in the Mathematica file [21]. While the specific numbers used may appear to be somewhat arbitrary (see also the discussion in Remark 3.11) they have been chosen to be used later in Theorem 4.6 and Theorem 4.9.…”

“…Proof. In the Mathematica file [21] we check, using interval arithmetic, that ρ ≥ C( 0 , r) and the radii polynomials P ( 0 , r, ρ) are negative for the choices (a) and (b). The result then follows from Corollary 3.9.…”

“…Let us consider B (r, ρ), which is centered atx (see Definition 2.10) with r and ρ taken as in Proposition 3.10(a). In the Mathematica file [21] we check that the following inequalities are satisfied:…”

“…In the Mathematica file [21] we check that (4.11) is satisfied for 0 = 0.1. Sinceᾱ 0 ≥ π 2 + 7.4247 × 10 −3 , and taking into account the control provided by Proposition 3.15 on the distance betweenα andᾱ in terms ofř α , we find thatα 0 ≥ᾱ 0 − 2 0řα ≥ π 2 + 6.830 × 10 −3 .…”