Intersection-number operators for curves on discs and Chebyshev polynomials
by
Stephen Humphries
Brigham Young University

We prove the folowing result and show a connection with intersection-number functions of curves on punctured discs:

Let R be an associative ring with identity and fix r in Z(R), where Z(R) is the centre of R. Define polynomials p_n=p_n(r) recursively by p₀ = -2, p₁ = r, p_n = -(rp_n-1 + p_n-2). Let f be a ring homomorphism and assume that f(r) = r. Define operators A_n = A_n(f, r), n >= 0, by A₀ = f - 1 and for n >= 0, we let A_n = f² + p_nf +1. Define operators B_n = B_n(f, r), n >= 0 by B_n = A₀ A₁ ... A_n.

Suppose that u, v in R and that B_n(u) = B_m(v) = 0 for n, m >= 0. Then B_n+m(uv) = 0.

Date received: January 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-21.