Lucas' theorem refined^∗

^∗ Dedicated to John B. Conway, the thesis advisor of the second author and the mathematical grandfather of the first, on his 60th birthday

Abstract

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a ₁,…,a _n+1 all having modulus one and φis the Blaschke product whose zeros are those of the derivative p ¹, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n+ l)-gon a ₁…a _n+1 with tangent points the midpoints of then+ l sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n= 2.

Keywords:

• Compression of the shift
• numerical range
• dilation
• AMS Subject Classification: 15A60, 47A12

^∗ Dedicated to John B. Conway, the thesis advisor of the second author and the mathematical grandfather of the first, on his 60th birthday

^∗ Dedicated to John B. Conway, the thesis advisor of the second author and the mathematical grandfather of the first, on his 60th birthday

Notes

^∗ Dedicated to John B. Conway, the thesis advisor of the second author and the mathematical grandfather of the first, on his 60th birthday