John Hilton Grace

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

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. 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.

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

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

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

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
 * $$a(z)=a_0+\tbinom{n}{1}a_1 z+\tbinom{n}{2}a_2 z^2+\dots+a_n z^n$$,


 * $$b(z)=b_0+\tbinom{n}{1}b_1 z+\tbinom{n}{2}b_2 z^2+\dots+b_n z^n$$

are two polynomials that satisfy the apolarity condition, i.e. $$a_0 b_n - \tbinom{n}{1}a_1 b_{n-1} + \tbinom{n}{2}a_2 b_{n-2} - \cdots +(-1)^n a_n b_0 = 0$$, then every neighbourhood that includes all zeros of one polynomial also includes at least one zero of the other.

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