John Hilton Grace

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^nare 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

*cite journal|last=Grace|first=J. H.|year=1936|month=January|title=The actual irreducibility of some finite systems of invariant forms|journal=Journal of the London Mathematical Society|volume=s1-11|pages=20–21|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-11/1/20|id=Zbl|0013.14601, JFM|62.0075.02|doi=10.1112/jlms/s1-11.1.20
*cite journal|last=Grace|first=J. H.|year=1930|month=April|title=Two contract theorems|journal=Journal of the London Mathematical Society|volume=s1-5|pages=121–124|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-5/2/121|doi=10.1112/jlms/s1-5.2.121
*cite journal|last=Grace|first=J. H.|year=1930|month=January|title=The algebraic expression of projective properties|journal=Journal of the London Mathematical Society|volume=s1-5|pages=62–67|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-5/1/62|doi=10.1112/jlms/s1-5.1.62
*cite journal|last=Grace|first=J. H.|year=1928|title=Binary and ternary forms with prescribed polar systems|journal=Proceedings of the London Mathematical Society|volume=s2-28|pages=421–430|url=http://plms.oxfordjournals.org/cgi/reprint/s2-28/1/421|id=JFM|54.0133.03|doi=10.1112/plms/s2-28.1.421
*cite journal|last=Grace|first=J. H.|year=1928|month=January|title=Surfaces related to a rational normal curve|journal=Journal of the London Mathematical Society|volume=s1-3|pages=34–38|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-3/1/34|doi=10.1112/jlms/s1-3.1.34
*cite journal|last=Grace|first=J. H.|year=1927|month=July|title=Note on ternary forms|journal=Journal of the London Mathematical Society|volume=s1-2|pages=182–185|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-2/3/182|doi=10.1112/jlms/s1-2.3.182
*cite journal|last=Grace|first=J. H.|year=1927|month=January|title=The four square theorem|journal=Journal of the London Mathematical Society|volume=s1-2|pages=3–8|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-2/1/3|doi=10.1112/jlms/s1-2.1.3
*cite journal|last=Grace|first=J. H.|year=1926|month=July|title=A point in enumerative geometry|journal=Journal of the London Mathematical Society|volume=s1-1|pages=167–170|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-1/3/167|doi=10.1112/jlms/s1-1.3.167
*cite journal|last=Grace|first=J. H.|year=1918|title=Note on a Diophantine approximation|journal=Proceedings of the London Mathematical Society|volume=s2-17|pages=316–319|url=http://plms.oxfordjournals.org/cgi/reprint/s2-17/1/316|doi=10.1112/plms/s2-17.1.316
*cite journal|last=Grace|first=J. H.|year=1918|title=Tetrahedra in relation to spheres and quadrics|journal=Proceedings of the London Mathematical Society|volume=s2-17|pages=259–271|url=http://plms.oxfordjournals.org/cgi/reprint/s2-17/1/259|id=JFM|47.0612.01|doi=10.1112/plms/s2-17.1.259
*cite journal|last=Grace|first=J. H.|year=1918|title=The classification of rational approximations|journal=Proceedings of the London Mathematical Society|volume=s2-17|pages=247–258|id=JFM|47.0166.01|url=http://plms.oxfordjournals.org/cgi/reprint/s2-17/1/247|doi=10.1112/plms/s2-17.1.247
*cite journal|last=Grace|first=J. H.|year=1904|title=Note on the foregoing paper|journal=Proceedings of the London Mathematical Society|volume=s2-1|pages=208–209|url=http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/208|doi=10.1112/plms/s2-1.1.208
*cite journal|last=Grace|first=J. H.|year=1904|title=Extension of two theorems on covariants|journal=Proceedings of the London Mathematical Society|volume=s2-1|pages=151–153|url=http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/151|id=JFM|34.0120.03|doi=10.1112/plms/s2-1.1.151
*cite book|last=Grace|first=J. H.|coauthors=A. Young|title=The Algebra of Invariants|publisher=Cambridge University Press|year=1903|url=http://books.google.com/books?id=TrAAAAAAMAAJ
*cite journal|last=Grace|first=J. H.|title=On the zeros of a polynomial|journal=Proceedings of the Cambridge Philosophical Society|volume=11|pages=352–357
*cite journal|last=Grace|first=J. H.|year=1902|month=May|title=On perpetuants|journal=Proceedings of the London Mathematical Society|volume=s1-35|pages=319–331|url=http://plms.oxfordjournals.org/cgi/reprint/s1-35/1/319|doi=10.1112/plms/s1-35.1.319
*cite journal|last=Grace|first=J. H.|year=1902|month=May|title=Types of perpetuants|journal=Proceedings of the London Mathematical Society|volume=s1-35|pages=107–114|url=http://plms.oxfordjournals.org/cgi/reprint/s1-35/1/107|doi=10.1112/plms/s1-35.1.107
*cite journal|last=Grace|first=J. H.|date=March 4 1901|title=A theorem on curves in a linear complex|journal=Proceedings of the Cambridge Philosophical Society|volume=11|pages=132–133|url=http://books.google.com/books?pg=PA132&id=IJ81AAAAIAAJ
*cite journal|last=Grace|first=J. H.|year=1901|month=March|title=Linear null systems of binary forms|journal=Proceedings of the London Mathematical Society|volume=s1-34|pages=168–172|url=http://plms.oxfordjournals.org/cgi/reprint/s1-34/1/168|id=JFM|33.0126.02|doi=10.1112/plms/s1-34.1.168
*cite journal|last=Grace|first=J. H.|year=1900|month=November|title=On a class of plane curves|journal=Proceedings of the London Mathematical Society|volume=s1-33|pages=193–196|url=http://plms.oxfordjournals.org/cgi/reprint/s1-33/1/193|doi=10.1112/plms/s1-33.1.193
*cite journal|last=Grace|first=J. H.|year=1898|title=Circles, spheres, and linear complexes|journal=Transactions of the Cambridge Philosophical Society|volume=16|pages=153–190

References

*cite journal|author=Hörmander, Lars|authorlink=Lars Hörmander|title=On a theorem of Grace|journal=Mathematica Scandinavica|volume=2|year=1954|pages=55–64|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002342987
*cite journal|author=Todd, John A.|authorlink=John Arthur Todd|title=John Hilton Grace|journal=Biographical Memoirs of Fellows of the Royal Society|volume=4|year=1958|month=November|pages=92–97|url=http://links.jstor.org/sici?sici=0080-4606(195811)4%3C92%3AJHG1%3E2|doi=10.1098/rsbm.1958.0008

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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