On univalent polynomials
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- by J. R. Quine PDF
- Proc. Amer. Math. Soc. 57 (1976), 75-78 Request permission
Abstract:
We define ${V_n} \subseteq {{\mathbf {C}}^{n - 1}}$ to be the set of $(n - 1)$-tuples $({a_2}, \ldots ,{a_n})$ such that the polynomial $p(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n}$ is univalent, i.e., one-to-one in $|z| < 1$. In this paper we construct a real polynomial $h$ of degree $4(2{(n - 1)^2} - 1)(n - 1)$ such that if $({a_2}, \ldots ,{a_n})$ is in the boundary of ${V_n}$ then $h(\operatorname {Re} {a_2},\operatorname {Im} {a_2}, \ldots ,\operatorname {Re} {a_n},\operatorname {Im} {a_n}) = 0$. This shows that the boundary of ${V_n}$ is a subset of an algebraic submanifold of ${R^{2(n - 1)}}$.References
- D. A. Brannan, Coefficient regions for univalent polynomials of small degree, Mathematika 14 (1967), 165–169. MR 220919, DOI 10.1112/S0025579300003764
- Miloš Kössler, Simple polynomials, Czechoslovak Math. J. 1(76) (1951), 5–15 = Čehoslovack. Mat. Ž. 1(76), 5–17 (1951). MR 47198
- J. R. Quine, On the self-intersections of the image of the unit circle under a polynomial mapping, Proc. Amer. Math. Soc. 39 (1973), 135–140. MR 313485, DOI 10.1090/S0002-9939-1973-0313485-X B. L. van der Waerden, Modern algebra. Vol. I, Springer, Berlin, 1930; English transl., Ungar, New York, 1949. MR 10, 587.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 57 (1976), 75-78
- MSC: Primary 30A34; Secondary 30A06
- DOI: https://doi.org/10.1090/S0002-9939-1976-0402024-3
- MathSciNet review: 0402024