On the dimension of spaces of linear transformations satisfying rank conditions (Q1073127)

Let \(L({\mathbb{C}}^ n,{\mathbb{C}}^ m)\) be the space of linear maps from \({\mathbb{C}}^ n\) to \({\mathbb{C}}^ m\) and let \(\ell (k,n,m)\), respectively \(\ell '(k,n,m)\), be the dimension of the largest linear subspace V in \(L({\mathbb{C}}^ n,{\mathbb{C}}^ m)\) such that every nonzero map in V has rank k, respectively \(\geq k\). By defining the sets \(Z_{\ell}(\alpha)\), the truncated polynomial ring in \(\alpha\) \((\alpha^{\ell}=0)\) with integer coefficients and \(Z^*_{\ell}(\alpha)\), the subgroup of the multiplicative group whose elements have zeroth order coefficient equal to one and \(Z^ k_{\ell}(\alpha) = \{Z^*_{\ell}(\alpha) \cap \{\)polynomials of degree \(\leq k\}\},\) the author proves the following main result. If \(\ell (k,n,m)\geq 1\) then there exists \(\eta \in Z^*_{\ell}(\alpha)\) satisfying: \(\eta \in Z_{\ell}^{m- k}(\alpha)\), \(\eta^{-1}\in Z^ k_{\ell}(\alpha)\) and \((1+\alpha)^ n\eta \in Z_{\ell}^{n-k}(\alpha).\) In particular he shows that \(\ell(m,n,m)\) \([=\ell(m,m,n)] = n-m+1\), \(\ell(n-1,n,n)=\begin{cases} 2 \text{ for n even} \\ 3\text{ for n odd} \end{cases}\) and \(\ell'(n-1,n,n)=4\).