This paper answers some questions left open in {\it V. Strassen}’s work [J. Reine Angew. Math. 384, 102-152 (1988; Zbl 0631.68033) and ibid. 413, 127-180 (1991)]. We show that the degeneration order $\trianglelefteq$ for bilinear maps cannot be extended from the set ${\frak B}\sp+$ of equivalence classes of bilinear maps to the Grothendieck ring ${\frak B}$ in a way compatible with the ring operations. The upper and lower support functionals $ζ\spθ,ζ\sbθ:{\frak B}\sp+\to {\bbfR}$ yield points in the asymptotic spectrum associated with a set of bilinear maps, thus giving necessary conditions for degeneration. We prove that the value of the lower support functional $ζ\sbθ$ of a generic bilinear map of n-dimensional spaces is $n\sp{1-\min θ\sb i+o(1)}(n\to \infty)$. We further show that the lower support functional is not additive - this in contrast to the upper support functional $ζ\spθ$.
Reviewer: P.Bürgisser