Determining consistency of a general overdetermined linear system For $m > 2$, consider the $m \times 2$ (overdetermined) linear system
$$A \mathbf{x} = \mathbf{b}$$
with (general) coefficients in a field $\mathbb{F}$; in components we write the system as
$$\left(\begin{array}{cc}a_{11} & a_{12} \\ \vdots & \vdots \\ a_{m1} & a_{m2} \end{array}\right)
\left(\begin{array}{c}x_1 \\ x_2\end{array}\right)
=
\left(\begin{array}{c}b_1 \\ \vdots \\ b_n\end{array}\right),$$
where $m > 2$, so that the system is overdetermined.
If $m = 3$ and the system is consistent, (equivalently, $\mathbf{b}$ is in the column space of $A$), then the columns of the augmented matrix $\pmatrix{ A \mid {\bf b}}$ are linearly dependent, and so
$$\det \pmatrix{ A \mid {\bf b}} = 0.$$
In particular, we have produced a polynomial in the components $(a_{ij}, b_j)$ of the linear system for which vanishing is a necessary condition for system's consistency. I'll call such polynomials polynomial obstructions for the system.
If $m > 3$, then we can produce ${m}\choose{3}$ such polynomials by considering the determinants of the $3 \times 3$ minors of $\pmatrix{ A \mid {\bf b}}$.
Are essentially all polynomials obstructions to the system essentially given by these, or are there others? Put more precisely: By definition the polynomial obstructions comprise an ideal in the polynomial ring $\mathbb{F}[a_{11}, \ldots, a_{m2}, b_1, \ldots b_m]$---do the determinants of the $3 \times 3$ minors generate this ideal? If not, how does one produce a complete set of generators?
More generally, for an $m \times n$ overdetermined linear system (so that $m > n$)
$$A \mathbf{x} = \mathbf{b},$$
we can produce polynomial obstructions by taking the determinants of the ${m}\choose{n+1}$ minors (of size $(n + 1) \times (n + 1)$). What are the answers to the obvious analogues to the above questions in the $n = 2$ case?

When $m=3$ and $\mathbb F$ is infinite, there are no other obstructions besides the determinant. When $\mathbb F$ is finite, there are many others : for example
if we put $\chi_{\mathbb F}(X)=\prod_{t\in {\mathbb F}^*} (X-t)$, $\chi(t)$ is zero iff 
$t$ is nonzero, so that the following $n$ polynomials are all obstructions : 
$$
w_i=b_i\prod_{j=1}^n\chi_{\mathbb F}(a_{ij})
$$
For the infinite case, one can use the following lemma :
Generalized Euclidean division. Let $A$ and $B$ be two polynomials
in ${\mathbb F}[X_1,X_2,\ldots,X_n,Y]$. Let $a={\sf deg}_Y(A)$, 
$b={\sf deg}_Y(B)$, and let $L$ be the leading coefficient of $B$
with respect to $Y$ (so that $L\in{\mathbb F}[X_1,X_2,\ldots,X_n]$
and $B-LY^b$ has degree $<b$ in $Y$). Then if $a \geq b$,
there are two polynomials $Q,R\in {\mathbb F}[X_1,X_2,\ldots,X_n,Y]$ such 
that $L^{a-b+1}A=QB+R$ and ${\sf deg}_Y(R)<b$.
Proof. Let ${\mathbb K}={\mathbb F}(X_1,X_2,\ldots,X_n)$. We can view
$A$ and $B$ as members of ${\mathbb K}[Y]$, and perform ordinary euclidian
division ; this yields $Q^{\sharp},R^{\sharp}\in {\mathbb K}[Y]$ such that
$A=Q^{\sharp}B+R^{\sharp}$. Since the division process involves
$a-b+1$ divisions by $L$, we see that $Q^{\sharp}$ and $R^{\sharp}$ are of
the form $\frac{Q}{L^{a-b+1}}$ and $\frac{R}{L^{a-b+1}}$ with 
$Q,R\in {\mathbb F}[X_1,X_2,\ldots,X_n,Y]$. This concludes the proof of the lemma.
Let us now explain how this can be used when $m=3$. Let $I$ be the ideal 
(in the ring ${\mathfrak R}={\mathbb F}(A_{11},A_{12},A_{13},A_{21},A_{22},A_{23},B_1,B_2,B_3)$ of all obstructions. In particular, the determinant
$$
\Delta=(A_{12}A_{23}-A_{13}A_{22})B_1+
(A_{13}A_{21}-A_{11}A_{23})B_2+
(A_{11}A_{22}-A_{12}A_{21})B_3 \tag{1}
$$
is a member of $I$. Let $P\in I$, and let $p={\sf deg}_{B_3}(P)$. By the
generalized Euclidean division property above, there are polynomials
in $Q,R$ in $\mathfrak R$ such that $(A_{11}A_{22}-A_{12}A_{21})^p P=\Delta Q+R$,
such that $R$ does not contain the variable $B_3$ (note that we need $p\geq 1$ in order to apply the lemma ; but if $p=0$, we can simply take $Q=0,R=P$). Then $R\in I$. Consider the set
$$
W=\bigg\lbrace (a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},b_1,b_2) \in 
{\mathbb F}^{8} \ \bigg| \ a_{11}a_{22}-a_{12}a_{21} \neq 0\bigg\rbrace \tag{2}
$$
Since $\mathbb F$ is infinite, $W$ is a Zariski-dense open subset of ${\mathbb F}^{8}$.
We have a natural map $\phi : W \to V(I)$, defined by
$$
\phi(a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},b_1,b_2)=
\bigg(a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},b_1,b_2,
-\frac{(a_{12}a_{23}-a_{13}a_{22})b_1+
(a_{13}a_{21}-a_{11}a_{23})b_2}{a_{11}a_{22}-a_{12}a_{21}}\bigg) \tag{3}
$$
For any $w\in W$, we have $R(\phi(w))=0$ since $R\in I$. We deduce $R(w)=0$
for all $w\in W$. Since $W$ is Zariski-dense, $R$ is zero eveywhere. So $R$ must be
the zero polynomial, $(A_{11}A_{22}-A_{12}A_{21})^p P=\Delta Q$. Since $A_{11}A_{22}-A_{12}A_{21}$
and $\Delta$ have no common factors, we see that $\Delta$ divides $P$.
