The linear dependence problem for power linear maps
The homogeneous dependence problem HDP(\(n,d\)) is related to the Jacobian conjecture and is as follows. If \(H=(H_{1},\dots,H_{n})\) is a homogeneous polynomial map of degree \(d\geq2\) from \(\mathbb{C}^{n}\) into itself, and the Jacobian \(JH\) is nilpotent, are \(H_{1},\dots,H_{n}\) linearly dependent over \(\mathbb{C}\)? Equivalently: are the rows of \(JH\) linearly dependent? A considerable amount is known about this problem [see \textit{M. de Bondt}, Proc. Am. Math. Soc. 134, No. 10, 2849--2856 (2006; Zbl 1107.14054)]. For example, the answer is yes when \(n\leq3\) for all \(d\) and when \(n=4\) and \(d=2\) or \(3\), but it is negative (for some \(d\)) when \(n\geq5\) and in particular when \(d=3\) and \(n\geq10\). The present paper addresses two open problems: Does HDP(\(n,2\)) have an affirmative answer for all \(n>4\), and does HDP(\(n,d\)) have an affirmative answer when \(H\) is a power map (that is, each component \(H_{i}\) is the \(d\)th power of a linear form)? Let \(A\) be an \(n\times n\) matrix, \(x\) a \(n\)-column vector of indeterminates, and let \((Ax)^{(k)}\) denote the \(k\)th Hadamard power of \(Ax\) (each entry raised to the \(k\)th power). Among other results, the authors show that the components of \(H(x)=(Ax)^{(d)}\) are linearly dependent if and only if \(\det(AA^{\ast})^{(d)}=0\) (Corollary 5), whilst if the entries in \((Ax)^{(d)}\) are linearly independent, then so are the entries of \((Ax)^{(k)}\) for all \(k\geq d\) (Corollary 7). This leads to various counterexamples for HDP(\(n,d\)) for power maps. They also show that if \(d\geq2\) is fixed then, in order to prove that HDP(\(n,d\)) has an affirmative answer for all \(n\geq1\), it is enough to prove the result for power maps (Corollary 15). Hence to decide whether HDP(\(n,2\)) has an affirmative answer for all \(n>4\) it is sufficient to decide whether this is true for power maps of degree \(2\).
scientific article; zbMATH DE number 5190972
scientific article