John Hilton Grace

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John Hilton Grace (1873 May 21-1958) was a British mathematician.

Contents

[edit] Theorem on zeros of a polynomial

If

a(z)=a_0+\tbinom{n}{1}a_1 z+\tbinom{n}{2}a_2 z^2+\dots+a_n z^n,
b(z)=b_0+\tbinom{n}{1}b_1 z+\tbinom{n}{2}b_2 z^2+\dots+b_n z^n

are two polynomials that satisfy the apolarity condition, ie a_0 b_n - \tbinom{n}{1}a_1 b_{n-1} + \tbinom{n}{2}a_2 b_{n-2} - \ldots +(-1)^n a_n b_0 = 0, then every neighborhood that includes all zeros of one polynomial also includes at least one zero of the other.[1][2]

[edit] Corollary

Let a(z) and b(z) be defined as in the above theorem. If the zeros of both polynomials lie in the unit disk, then the zeros of the "composition" of the two, c(z)=a_0 b_0 + \tbinom{n}{1}a_1 b_1 z + \tbinom{n}{2}a_2 b_2 z^2 + \ldots + a_n b_n z^n, also lie in the unit disk.[1]

[edit] Publications

[edit] References

[edit] Footnotes

  1. ^ a b Szegő, Gábor (1922). "Bemerkungen zu einem Satz von J H Grace über die Wurzeln algebraischer Gleichungen". Mathematische Zeitschrift 13: 28–55. doi:10.1007/BF01485280.  (German)
  2. ^ Rahman, Qazi I.; Gerhard Schmeisser (2002). "Grace's theorem and equivalent forms", Analytic Theory of Polynomials. Oxford University Press, 107. ISBN 0198534930. 
Persondata
NAME Grace, John Hilton
ALTERNATIVE NAMES
SHORT DESCRIPTION British mathematician
DATE OF BIRTH 1873 May 21
PLACE OF BIRTH
DATE OF DEATH 1958
PLACE OF DEATH
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