John Hilton Grace

John Hilton Grace
Born 21 May 1873(1873-05-21)
Halewood, Lancashire
Died 4 March 1958(1958-03-04) (aged 84)
Nationality  British
Fields Mathematics

John Hilton Grace (21 May 1873 - 4 March 1958) was a British mathematician.

Contents

Early life

He was born in Halewood, near Liverpool, the eldest of the six children of a farmer. He was educated at the village school and the Liverpool Institute. From there in 1892 he went up to Peterhouse College, Cambridge to study mathematics.[1]

Career

He was made a Fellow of Peterhouse in 1897 and became a Lecturer of Mathematics at Peterhouse and Pembroke colleges. An example of his work was his 1902 paper on the The Zeros of a Polynomial. In 1903 he collaborated with Alfred Young on their book Algebra of Invariants.[1]

He was elected a Fellow of the Royal Society in 1908.[1]

He spent 1916-1917 as Visiting Professor in Lahore and deputised for Professor MacDonald at Aberdeen University during the latter part of the war.[2]

In 1922 a breakdown in health forced his retirement from academic life and he spent the next part of his life in Norfolk.[1]

He died in Huntingdon in 1958 and was buried in the family grave at St. Nicholas Church, Halewood.

Theorem on zeros of a polynomial

If

a(z)=a_0%2B\tbinom{n}{1}a_1 z%2B\tbinom{n}{2}a_2 z^2%2B\dots%2Ba_n z^n,
b(z)=b_0%2B\tbinom{n}{1}b_1 z%2B\tbinom{n}{2}b_2 z^2%2B\dots%2Bb_n z^n

are two polynomials that satisfy the apolarity condition, i.e. a_0 b_n - \tbinom{n}{1}a_1 b_{n-1} %2B \tbinom{n}{2}a_2 b_{n-2} - \ldots %2B(-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.[3][4]

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 %2B \tbinom{n}{1}a_1 b_1 z %2B \tbinom{n}{2}a_2 b_2 z^2 %2B \ldots %2B a_n b_n z^n, also lie in the unit disk.[3]

Publications

References

  1. ^ a b c d Todd, J. A. (1958). "John Hilton Grace. 1873-1958". Biographical Memoirs of Fellows of the Royal Society 4: 92–97. doi:10.1098/rsbm.1958.0008. 
  2. ^ Todd, J. A. (1959). "John Hilton Grace". Journal of the London Mathematical Society: 113–117. doi:10.1112/jlms/s1-34.1.113. 
  3. ^ 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. http://resolver.sub.uni-goettingen.de/purl?GDZPPN002366630.  (German)
  4. ^ Rahman, Qazi I.; Gerhard Schmeisser (2002). "Grace's theorem and equivalent forms". Analytic Theory of Polynomials. Oxford University Press. pp. 107. ISBN 0198534930. 

Further reading

External links