The Bourbaki degree of the syzygy module of 2 $\times$ 4 matrices

We introduce and study the Bourbaki degree as a numerical invariant for $2\times 4$ matrices $\mathrm{\Theta }$ of homogeneous polynomials over a polynomial ring $R=k[x_1,\dots ,x_n]$. This invariant, defined via a Bourbaki sequence for the syzygy module $\mathrm{Syz}(\mathrm{\Theta })$, generalizes previous constructions for plane curves and Jacobian matrices. Our main result is an explicit formula expressing the Bourbaki degree in terms of the degrees of the rows, the initial degree of a syzygy, and the first two Hilbert coefficients of the cokernel module $\mathcal{Q}=\mathrm{coker}(\mathrm{\Theta })$. We apply this framework to two important cases. First, matrices with constant first row, which are determined by a three-equigenerated ideal $J=(f_1,f_2,f_3)$, where we show the Bourbaki degree measures how far $J$ is from being a perfect ideal, and we completely characterize its smaller and larger values. Second, for a linear matrix, we use the Kronecker–Weierstrass classification to determine all possible Bourbaki degrees and homological types. This classification reveals the existence of a linear matrix with Bourbaki degree equal to 2, a value that does not occur for Jacobian matrices. Finally, in the geometric context of ${\mathbb{P}}^3$, we provide a sufficient condition for $\mathrm{Syz}(\mathrm{\Theta })$ to define a codimension one distribution and obtain bounds on the Bourbaki degree when the initial degree is small.