= Complex bilinear dimension bound
{title2=$\dim\operatorname{im}\phi\geq\dim U+\dim V-1$}
For nonzero finite-dimensional complex <vector spaces> $U,V$, suppose a <bilinear map> never vanishes on a pair of nonzero vectors. Equivalently, each map obtained by fixing one nonzero argument is injective. If its associated map on $U\otimes V$ has image dimension $r$, projectivization gives a map to $\mathbb{CP}^{r-1}$ whose positive degree-two class pulls back to $x+y$. The nonzero top power $(x+y)^{\dim U+\dim V-2}$ in the product <cohomology ring of complex projective space> proves the bound. Multiplication of polynomials of bounded degree attains equality. Nonzero hypotheses matter: with a zero factor, slice conditions can be vacuous.
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