John Hilton Grace

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

Theorem on zeros of a polynomial

If: $a(z)=a_0+ binom{n}{1}a_1 z+ binom{n}{2}a_2 z^2+dots+a_n z^n$ ,

: $b(z)=b_0+ binom{n}{1}b_1 z+ binom{n}{2}b_2 z^2+dots+b_n z^n$ are two polynomials that satisfy the apolarity condition, ie $a_0 b_n - binom{n}{1}a_1 b_{n-1} + binom{n}{2}a_2 b_{n-2} - ldots +(-1)^n a_n b_0 = 0$ , then every neighborhood that includes all zeros of one polynomial also includes at least one zero of the other.cite journal|author=Szegő, Gábor|authorlink=Gábor Szegő|title=Bemerkungen zu einem Satz von J H Grace über die Wurzeln algebraischer Gleichungen|journal=Mathematische Zeitschrift|volume=13|year=1922|pages=28–55|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002366630|doi=10.1007/BF01485280 de icon] cite book| last=Rahman | first=Qazi I. | coauthors=Gerhard Schmeisser | year=2002 | chapter=Grace's theorem and equivalent forms | title=Analytic Theory of Polynomials | pages=107 | isbn=0198534930 | publisher=Oxford University Press]

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, $c(z)=a_0 b_0 + binom{n}{1}a_1 b_1 z + binom{n}{2}a_2 b_2 z^2 + ldots + a_n b_n z^n$ , also lie in the unit disk.

Publications

References

Footnotes

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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