Some conjectures on multivariate polynomial matrices (Q2702517)

The authors prove that several topics on multivariate polynomial matrices, like unimodular polynomial completion and several primeness and coprimeness issues, are related to a generalized Serre conjecture. More precisely, let \(F\) be an \(m\times l\) matrix with entries in \(K[z_1,z_2,\dots{},z_n]\), denoted by \(K[z]\), the set of polynomials in \(n\) variables over a field \(K\), and let \(a_1,a_2,\dots{},a_{\beta}\) denote the \(l \times l\) minors of the matrix \(F\). Extracting the greatest common divisor (g.c.d.) \(d\) of \(a_1,a_2,\dots,a_{\beta}\) gives \(a_i =d b_i \) for \( i=1,\dots{}, \beta\). Assume that \(b_1,\dots,b_{\beta}\) are zero coprime. Then the following conjectures are equivalent.NEWLINENEWLINENEWLINEConjecture 1. There exists \(E\in K^{l \times m} [z]\), such that \(U=[F E] \in K^{ m \times m }[z]\), with \(\det U= d\). (Generalized Serre conjecture)NEWLINENEWLINENEWLINEConjecture 2. \(F\) can be factored as \(F=F_0 G_0\) for some \( F_0 \in K^{ m \times l }[z]\), \(G_0 \in K^{l \times l} [z]\), \(\det G_0=d\).NEWLINENEWLINENEWLINEConjecture 3. There exists \(H\in K^{l \times m} [z]\) such that \(H F= G_0\), for some \(G_0 \in K^{l \times l} [z]\), with \(\det G_0=d\).NEWLINENEWLINENEWLINEConjecture 4. There exists \(B \in K^{(m-l)\times m} [z]\), with \(B\) being Zero Left Prime such that \(BF=0_{ (m-l) \times l}\).NEWLINENEWLINENEWLINEThe set of conjectures is complete, i.e., the solution of any one of them would solve the remaining ones. If the polynomials \(b_1,\dots,b_{\beta}\) are not zero coprime, the following conjecture is formulated: NEWLINENEWLINENEWLINEConjecture 5. Assume that \(d,b_1,\dots,b_{\beta}\) are zero coprime, then \(F\) can be factored as \(F=F_0 G_0\) for some \( F_0 \in K^{ m \times l }[z]\), \(G_0 \in K^{l \times l} [z]\), \(\det G_0=d\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00062].