linear algebra – Existence of a matrix with non-zero minors

Is there a matrix $X$ of size $n times k$ in the field $mathbb{F}_2$, such that any $k$ rows of the following matrix $M$ have full rank:
$$
M = left(begin{matrix} X\
I_kend{matrix} right),
$$

where $I_k$ is the identity matrix of size $k times k$ in the field $mathbb{F}_2$.

I guess that $n$ will need to be sufficiently large compared to $k$. I know that there is a connection between minors and exterior products but I am not sure if it is any useful.

co.combinatorics – Equivalence of induced minors and induced topological minors

Let $G, H$ be two graphs. It is easy to see that if $G$ contains $H$ as a topological minor, then $G$ contains $H$ as a minor (I will append definitions to the end of the question). In the other direction, it is not difficult (see e.g. Proposition 1.7.3 of Reinhard Diestel’s book ‘Graph Theory’, 5th ed) to show that the converse is true if the maximum degree of $H$ is at most 3. In fact, the converse is true for all graphs $G$ if and only if $H$ has maximum degree at most 3.

What about the induced variants of minors and topological minors? Again, it is trivial that every induced topological minor is also an induced minor. For the converse, the graph $H$ certainly needs to be subcubic (max-degree at most 3). But the converse is not true for all subcubic graphs: attach a degree-1 vertex to each vertex of a triangle. This graph contains the claw $K_{1,3}$ as an induced minor – just contract the triangle – but not as an induced topological minor (because it does not contain the claw as an induced subgraph).

So the question is: for which graphs $H$ is the following true?

(***) Every graph that contains $H$ as an induced minor also contains $H$ as an induced topological minor

Some initial thoughts:
Extending the above construction, we can derive some necessary conditions on such graphs:

  • given $H$, obtain $G$ from $H$ by replacing every $v in V(H)$ by a clique $K_v$ of order $d_H(v)$ (the degree of $v$ in $H$) and join, for each edge $uv in E(H)$, some $x in K_u$ to some $y in K_v$, in a way so that every $x in V(G)$ has exactly one neighbour outside the clique $K_v ni x$.
  • Then $H$ is an induced minor of $G$: just contract each clique $K_v$ to a single vertex.
  • $G$ does not contain the claw $K_{1,3}$ as an induced topological minor: for every induced subgraph $G’ subseteq G$, every degree-3 vertex of $G’$ has at least two adjacent neighbors.
  • $G$ does not contain $K_4$ as an induced topological minor: suppose $G’ subseteq G$ was an induced subdivision of $K_4$ and let $a_1, a_2, a_3, a_4 in V(G’)$ be the four vertices of degree-3. Let $v in V(H)$ with $a_1 in K_v$. Then two of the three neighbours of $a_1$ must again lie in $K_v$ (because $a_1$ has at most one neighbour outside), making them adjacent. Therefore, they must actually be among $a_1, a_2, a_3, a_4$, wlog $a_2, a_3 in K_v$. But then apply the same argument to $a_4 in K_w$ for some $w neq v$ – a contradiction.

Therefore, if $H$ contains $K_4$ of $K_{1,3}$ as an induced topological minor, then $H$ cannot have the propery (***) (by transitivity of the ‘induced topological minor’ relation).

Now the missing definitions. Let $G$ and $H$ be graphs.

  • $H$ is a minor of $G$ if $H$ can be obtained from $G$ by deleting vertices and edges and contracting edges.
  • $H$ is an induced minor of $G$ if $H$ can be obtained from $G$ by deleting vertices and contracting edges.
  • $H$ is a topological minor of $G$ if $G$ contains a subgraph which is a subdivision of $H$, i.e. it can be obtained from $H$ by subdividing edges.
  • $H$ is an induced topological minor of $G$ if $G$ contains an induced subgraph which is a subdivision of $H$.

linear algebra – What are the determinants of minors of a “lower triangular” matrix? (tensor product)

I’m interested in whether linear maps are surjective, so I need to find determinants of minors of matrices. $k$ is some field, perhaps $mathbb{R}$ or $mathbb{C}$.

For one example, $k^n to k^n$ was lower triangular, so the determinant is the product along the diagonal.

The next example has a bilinear map $k^n times k^n to k^n$, i.e. a linear map $k^{n^2} = k^n otimes k^n to k^n$. It is also “lower triangular” in the following sense: write coordinates as $(bigoplus_{i=1}^n k cdot t_i) otimes (bigoplus_{j=1}^n k cdot u_j) to bigoplus_{p=1}^n k cdot epsilon_p$. Then the coefficient of $epsilon_p$ in the image of $t_i otimes u_j$ is zero if $i + j > p$ (“upper simplicial?”). Is there a similarly succinct way to say what determinants of minors of this matrix look like? What about if I take higher tensor products?

This is related to the tensor product of matrices, except it’s not the tensor product of two maps, but a bilinear map.

reference request – Can all matroids representable over a fixed finite field having packed broken circuits be characterised by a finite set of forbidden minors?

For a matroid $M$ grounded on $E$ and $ein E$ let $smallmathcal{C}_e(M)={Xsubseteq Esetminus {e}:Xcup {e}text{ is a circuit of }M}$ and refer to elements of $mathcal{C}_e(C)$ as broken circuits of $e$ also let $M^*$ be the dual matroid of $M$ and call $M$ a packed matroid if and only if we have:

$$smallforall ein E(max{|mathcal{P}|:mathcal{P}subseteq mathcal{C}_e(M)text{ is a pairwise disjoint family of sets}}=min{|S|:Sin mathcal{C}_e(M^*)})$$

Now with a little bit of work one can prove that packed matroids are minor closed and thus $M$ is a packed matroid iff $M$ has no minimal non-packed matroid minor. Now with all of that said my question is this: for which finite fields $mathbb{F}$ can the packed matroids represented over $mathbb{F}$ be characterised by a finite set of forbidden minors? E.g. Is it true for every finite field $mathbb{F}$?


The reason I ask this (and think it is an interesting question) is because were it true, and were someone to list those forbidden minors for packed matroids over some finite field then I think one could generalize several linear programming algorithms used in network theory that take advantage of
Menger’s theorem theorem. This is because for a graphic matroid $M(G)$ the ground set is $E(G)$ and the broken circuits of any $uvin E(G)$ in $M(G)$ are the $u$$v$ paths of $G-uv$ while the broken cocircuits of any $uvin E(G)$ in $M(G)$ are the $u$$v$ edge separators of $G-uv$ i.e. Menger’s theorem is equivalent to the statement that all graphic matroids are packed.

schengen – Long stay France visa for minors

My wife and I have Long Stay Visa in the “entrepreneur / profession libérale” category from France, and already rented a house in Paris to live.

Two weeks ago our baby girl was born in Canada, and now we are planning to return to Paris in 6 weeks, I was wondering should we request a visa for our newborn baby? Or we should just go to Paris (She needs no visa for entry and staying for 90 days) or what?

I am not sure what should we do, Please let me know if anyone knows what to do.

usa – Exiting/entering the US – minor’s US passport expired

I’m a US citizen as is my son. Due to COVID-19, I was unable to renew my son’s passport. We have tickets and are supposed to travel to Spain within the next two weeks. He also has a current Spanish passport (holds dual citizenship). Will I have trouble boarding the plane or leaving/entering the US? I can provide a birth certificate, street address and school ID if needed. Will I need other documents, or can I get a letter from your agency to assist me with this endeavor?