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Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc, n\}$ with size $k$ let $e_I$ be the function which evaluates the determinant of the $k$ by $k$ submatrix found by taking only columns indexed by elements of $I$.

Let $A$ be a $k$ by $n$ matrix. It's well known that $$e_I(A) = 0 \text{ for all } I \subseteq [n], |I|=k \Rightarrow \operatorname{rank}(A)<k.$$ Moreoever, one can find an example of a matrix $A_I$ for which $e_J(A_I) \neq 0$ if and only if $I=J$, so no proper subset of $S:=\{e_I: I \subseteq [n], |I|=k\}$ suffices to check the rank of $A$.

If we allow linear combinations of the elements of $S$ we can sometimes get away with fewer functions. The smallest nontrivial example arises when $k=2$, $n=4$. There we have the Plücker relation $$e_{1,2}e_{3,4}-e_{1,3}e_{2,4}+e_{1,4}e_{2,3} = 0.$$ This implies that the vanishing of $$\{e_{1,2}-e_{3,4},e_{1,3},e_{2,4},e_{1,4},e_{2,3}\}$$ is sufficient to show a $2$ by $4$ matrix has rank $<2$.

Let's call a set of linear combinations of $e_I$ with the property above a "rank-detecting set", and write $\beta(k,n)$ for the size of the smallest rank detecting set for $k$ by $n$ matrices. Clearly $\beta(k,n) \leq \binom n k$. It's quite easy to show that $\beta(n-1,n) = n$, and the example above shows that $\beta(2,4) \leq 5$.

Have you come across this notion before? Is anything known in general about the numbers $\beta(k,n)$?

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  • $\begingroup$ TeX note: binomial coefficients should be typeset as $\binom n k$ \binom n k or, if you want a bigger version, $\dbinom n k$ \dbinom n k, not as $\begin{pmatrix} n \\ k \end{pmatrix}$ \begin{pmatrix} n \\ k \end{pmatrix}. $\endgroup$
    – LSpice
    Mar 7 at 14:33

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This notion is indeed well-known and connected to the arithmetical rank of the ideal $I_k$ of $C[V]$ generated by the maximal minors.

Short answer : that $\beta(k,n) = mk - k^2 +1$ and there is always a set of $mk-k^2+1$ linear combinations of maximal minors which is a rank detecting set (with your words).

Longer answer : Let $R$ be a commutative ring (say Noetherian) and $I$ a non trivial ideal of $R$. We definie the arithmetical rank of $I$ as: $$ \mathrm{ara}(I) : \mathrm{min} \{m \geq 1, \ \exists f_1, \ldots, f_n \in R, \ \textrm{such that} \ \sqrt{I} = \sqrt{(f_1, \ldots, f_m)} \}$$

In other words, the arithmetical rank of $I$ is the minimum number of equations sufficient to define, set-theoretically, the same variety as $I$ in $\mathrm{Spec}(R)$.

Let $V$ be the set of $n \times k$ matrices, $\mathbb{C}[V]$ the algebra of polynomial functions over $V$ and $t \leq \mathrm{min}(m,n)$. We denote by $I_t$ the ideal of $\mathbb{C}[V]$ generated by $t \times t$ minors of $\mathbb{C}[V]$. In The number of equations defining a determinantal variety, by Brüns and Schwanzl, it was proved that $\mathrm{ara}(I_t) \geq mk-t^2 +1$. It was furthermore proved before by Bruns, using the theory of algebras with straightening laws (see the references in the paper I gave a link to) that there is a set of linear combinations of minors of size $nk - t^2+1$ which generates an ideal having same ridical as $I_t$. These sets are studied in details in the chapter 5 of Determinantal Rings by Bruns and Vetter.

Note that the cas $t = \mathrm{min}(k,m)$, which seems to be the case of interest for you, has been settled before by Hochster (the precise reference is given in the fist chapter of the book by Bruns and Vetter, but I can't find it on top of my head).

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