tags:

(though wine may be needed for truth)

Worked out last night over a quiet pint:

\begin{aligned} & \quad (3x^2-2x^3)(3y^2-2y^3)-(3z^2-2z^3) \\ & = 6xy(x^2-x)(y^2-y) + (xy-z)^2(3-2xy-4z)+6(xy-z)(z-z^2) \end{aligned}

Consequently: if $x$, $y$ and $z$ lie in a commutative Banach algebra, and $x^2-x$, $y^2-y$, $z^2-z$ and $xy-z$ all have norm $O(\varepsilon)$, then the expression on the left hand side has norm at most $O(\varepsilon^2)$.

[Title corrected 2011-12-08]