# John Hilton Grace

John Hilton Grace
Born 21 May 1873
Halewood, Lancashire
Died 4 March 1958 (aged 84)
Nationality  British
Fields Mathematics
Known for Grace–Walsh–Szegő theorem
Notable awards Fellow of the Royal Society[1]

John Hilton Grace FRS[1] (21 May 1873 – 4 March 1958) was a British mathematician. The Grace–Walsh–Szegő theorem is named in part after him.[2]

## Early life

He was born in Halewood, near Liverpool, the eldest of the six children of farmer William Grace and Elizabeth Hilton. 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] His nephew, his younger sister's son, was the animal geneticist, Alan Robertson FRS.

## 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 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.[3]

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

${\displaystyle a(z)=a_{0}+{\tbinom {n}{1}}a_{1}z+{\tbinom {n}{2}}a_{2}z^{2}+\dots +a_{n}z^{n}}$,
${\displaystyle b(z)=b_{0}+{\tbinom {n}{1}}b_{1}z+{\tbinom {n}{2}}b_{2}z^{2}+\dots +b_{n}z^{n}}$

are two polynomials that satisfy the apolarity condition, i.e. ${\displaystyle a_{0}b_{n}-{\tbinom {n}{1}}a_{1}b_{n-1}+{\tbinom {n}{2}}a_{2}b_{n-2}-\cdots +(-1)^{n}a_{n}b_{0}=0}$, then every neighborhood that includes all zeros of one polynomial also includes at least one zero of the other.[4][5]

### Corollary

Let ${\displaystyle a(z)}$ and ${\displaystyle b(z)}$ 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, ${\displaystyle c(z)=a_{0}b_{0}+{\tbinom {n}{1}}a_{1}b_{1}z+{\tbinom {n}{2}}a_{2}b_{2}z^{2}+\cdots +a_{n}b_{n}z^{n}}$, also lie in the unit disk.[4]

## References

1. 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. JSTOR 769502.
2. ^ Hörmander, Lars (1954). "On a theorem of Grace". Mathematica Scandinavica. 2: 55–64. doi:10.7146/math.scand.a-10395.
3. ^ Todd, J. A. (1959). "John Hilton Grace". Journal of the London Mathematical Society: 113–117. doi:10.1112/jlms/s1-34.1.113.
4. ^ a b Szegő, Gábor (1922). "Bemerkungen zu einem Satz von J H Grace über die Wurzeln algebraischer Gleichungen". Mathematische Zeitschrift (in German). 13: 28–55. doi:10.1007/BF01485280.
5. ^ Rahman, Qazi I.; Gerhard Schmeisser (2002). "Grace's theorem and equivalent forms". Analytic Theory of Polynomials. Oxford University Press. p. 107. ISBN 0-19-853493-0.