A note on the Hopf-Stiefel function

We give a short elementary proof for a formula for the Hopf–Stiefel function that was found recently by Plagne via additive number theory. c © 2006 Elsevier Ltd. All rights reserved. The Hopf–Stiefel function ◦, introduced by Hopf [3] and Stiefel [8], plays an important role in additive number theory and bilinear algebra; see e.g. Plagne [6] for an overview and many references, for example [5] and [7]. It is defined on positive integral variables r and s as the smallest positive integer n such that (x + y)n is contained in the ideal (xr , ys) in the polynomial ring F2[x, y]. Putting it otherwise, r ◦ s = min { T | ( T j ) ≡ 0 (mod 2) for every T − s < j < r }