Vengeance! The Vultee Vengeance Dive Bomber by Peter C. Smith

By Peter C. Smith

The Vultee Vengeance Dive Bomber

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3 Zeros of Slice Regular Functions 41 where q ∈ [q1 ] for all = 2, . . , r and q unique factorization of P(q). +1 = q¯ , for = 1, . . , r − 1 is the Any quaternionic polynomial admits a factorization as described below. We do not prove this result here since we will prove a result in the more general framework of slice regular functions. We refer the reader to [112] for more details. 9 Let P(q) be a slice regular polynomial of degree m. Then there exist p, m1 , . . , mp ∈ N, and w1 , . . 12) where Q is a slice regular polynomial with coefficients in H having (at most) only nonspherical zeroes.

R H/n. Each ball Bi contains λi zeros of P and λ 1 ≥ λ2 ≥ . . ≥ λr . Note that, by construction, (λ1 + · · · + λr ) H = H. n The zeros of P contained in Bi will be said of rank λi . Step 4. Consider the balls Σ1 , . . , Σr with the same center as B1 , . . , Br , and with the radius of Σi which is twice the radius of Bi , i = 1, . . , r. Let q ∈ H\(Σ1 ∪. ∪Σr ). Consider the closed ball centered at q and with radius λ0 H/n for some λ0 ∈ N. By the construction above we have that the ball B0 intersects the balls Bi with radius at least equal to λ0 H/n.

2, point (3) shows that g n • w = (g • w) n and the equality follows. The general case follows by considering f as the sum of the summable family {q n an } and using the first part of the proof: ∞ ( f • g) • w = ∞ (g n=0 n • w)an = (g • w) n an = f • (g • w). n=0 The discussion, so far, was on formal power series without specifying the set of convergence. We now consider this aspect, by proving the following result which is classical for power series with coefficients in a commutative ring, see [37].

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