John Hilton Grace Explained

John Hilton Grace
Birth Date:	1873 5, df=yes
Birth Place:	Halewood, Lancashire
Nationality:	GBR
Fields:	Mathematics
Known For:	Grace–Walsh–Szegő theorem
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, 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

a(z)=a_{0+\tbinom{n}{1}a}₁z+\tbinom{n}{2}a₂

	2+...+a
z
	n

zⁿ

b(z)=b_{0+\tbinom{n}{1}b}₁z+\tbinom{n}{2}b₂

	2+...+b
z
	n

zⁿ

are two polynomials that satisfy the apolarity condition, i.e.

a₀b_n-\tbinom{n}{1}a₁b_n-1+\tbinom{n}{2}a₂b_n-2- … +(-1)ⁿa_nb₀=0

, then every neighbourhood that includes all zeros of one polynomial also includes at least one zero of the other.^[4] ^[5]

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)=a₀b₀+\tbinom{n}{1}a₁b₁z+\tbinom{n}{2}a₂b₂z²+ … +a_nb_nzⁿ

, also lie in the unit disk.

Publications

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

Notes and References

Todd . J. A. . J. A. Todd. 10.1098/rsbm.1958.0008 . John Hilton Grace 1873-1958 . . 4 . 92–97 . 1958 . 769502. 72982665 .
Hörmander, Erling. Erling Hörmander. On a theorem of Grace. Mathematica Scandinavica. 2. 1954. 55–64. 10.7146/math.scand.a-10395. free.
Todd . J. A. . John Hilton Grace . 10.1112/jlms/s1-34.1.113 . Journal of the London Mathematical Society . 113–117 . 1959 .
Szegő, Gábor. Gábor Szegő. Bemerkungen zu einem Satz von J H Grace über die Wurzeln algebraischer Gleichungen. Mathematische Zeitschrift. 13. 1922. 28–55. 10.1007/BF01485280. 121862267. de.
Book: Rahman, Qazi I. . Gerhard Schmeisser . 2002 . Grace's theorem and equivalent forms . Analytic Theory of Polynomials . 107 . 0-19-853493-0 . Oxford University Press.