Interesting (and not sufficiently answered) questions on math.SE

Question

In light of the generally positive response to this meta-thread, I am creating this to help advertise (and hopefully fully resolve) "interesting" questions on math.SE. I'm not really certain what the guidelines for this should be, but the following are likely uncontroversial:

1. Each answer should link directly to a single question on math.SE which has not been sufficiently resolved (and which is answerer believes would be of interest to members of the MO community).
2. Each answer should include a short summary of the question asked. There's no need to repost the math.SE question verbatim, but it should be fairly clear what the question is without following the link.
3. Feel free to downvote/vote to delete suggestions that aren't "up to snuff". Of course, commentary about why such questions aren't really of interest to the MO community would also be valuable.
4. Abusing this thread is an easy way to ensure it is closed/deleted. This thread will stay open only as long as the MO community feels it serves a useful purpose. In particular, don't post links to math.SE question that you know are open problems, and, at the other end of the spectrum, don't post links to math.SE questions that are essentially homework exercises.

(More active MO members should feel free to massage and expand these guidelines to better fit this community's expectations.)

Answer 1

Can someone figure out what is going on in this question about representation varieties of associative algebras? It looks like the OP may have found an error in a published article of Smalø, but I can't prove that its an error.

Answer 2

Okay, here's one that I think is interesting and has remained unanswered for over a month. (It might also test out this system.)

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

If it isn't clear from the title, this question (asked by Willie Wong) asks whether the inverse of any connected bijection $\mathbb{R}^n \to \mathbb{R}^n$ is itself connected (where a mapping $f : X \to Y$ between topological spaces — not necessarily continuous — is connected if $f[A]$ is connected for every connected $A \subseteq X$).

Answer 3

This question asks, given a homogenous degree $n$ polynomial map $f: \mathbb{Z}^n \to \mathbb{Z}$, how can we test if it is a norm map from a number field? I know roughly what the answer must be: Over $\mathbb{Q}^\text{alg}$, $f$ should factor as $n$ distinct hyperplanes and $f$ should be irreducible over $\mathbb{Q}$. That should make $f: \mathbb{Q}^n \to \mathbb{Q}$ into a norm up to scaling. But I can't quite figure out how to switch from $\mathbb{Q}$ to $\mathbb{Z}$, and the "up to scaling" is annoying.

I realize that "I know roughly what the answer should look like, but can't/don't-want-to work out the details" is not a good way to sell a problem. But it is a natural question and its been bugging me that is unanswered for so long. My hope is that, by advertising it here, someone will know a paper which already does all of this.

Answer 4

Does the open mapping theorem imply the Baire category theorem?

The open mapping theorem and the Baire category theorem are both theorems of ZFC, but are independent of ZF. Working in ZF, one can (apparently) prove that BCT implies OMT. Does the converse hold in ZF?

This is currently the highest-voted question with the functional-analysis tag.

Answer 5

Is there an injective cubic polynomial $\mathbb Z^2\rightarrow\mathbb Z$?

This question asks exactly what its title suggests: is there a polynomial of degree $3$ in $\mathbb Z[x,y]$ whose induced map $\mathbb Z^2\rightarrow\mathbb Z$ is injective? The question is trivial for polynomials of degree $4$ (where the answer is yes) or polynomials of degree $2$ or less (where the answer is no).

I recognize that this may not be wholly in the spirit of the thread, since I have suspicions that the question is open given the existence of two unanswered questions on MO asking about polynomial bijections (one on $\mathbb Q^2$ and one on $\mathbb Z^2$) and one about surjections. However, I suspect that my question is considerably easier, as it asks only about injectivity and only about cubics (where, from my amateur knowledge, it seems like more tools are available than for general polynomials). In any case, I note that there has been some amount of research regarding the linked questions, and think that it's not unlikely that an expert in the field could have some references or insights to contribute to the question.

Answer 6

Here is a question which seems interesting,

What does it take to divide by $2$?

In a nutshell, we know that in $\sf ZF$, assuming classical logic, we can prove that if $A\times\{0,1\}$ and $B\times\{0,1\}$ are equipotent, then $|A|=|B|$.

How far can this bend before it breaks? Can we do the same in intuitionistic set theories?

(Minor bump: the question currently carries a 500 points bounty.)

Answer 7

Weil Pairing on Linear Algebraic Groups

There is of course a Weil pairing on abelian varieties (it pairs an element of the torsion with something in the torsion of the dual) which are projective algebraic groups but does a similar construction exist for linear algebraic groups?

Answer 8

Would someone who knows cyclotomic fields, or algebraic number theory more generally, please take a look at $A_4$ extension of $\mathbb{Q}$ ramified at one prime? The question asserts that an $A_4$ extension of $\mathbb{Q}$ ramified at only one finite prime $p$ must be totally real.

I have made some partial progress, but I feel like someone who really knows cyclotomic fields and how to use number theory software should be able to kill this.

Answer 9

Another user suggested that I should advertise this question in MO also, so here comes. It is just a number theoretic curiosity spawned by an earlier question where small cases were handled by a computer search, asking whether to a given $t$ we can find a $k$ such that $(k!)^2\mid (2k-t)!$.

I will try and collect interesting stuff on Math.SE to the Pearl Dive — a pet project of yours truly and one of the moderators. The level of activity there took a dive, when the pandemic measures made me put more effort into producing material for my courses. If you see something you find "cute" ($\approx$ not straight out of several text books) in Math.SE, please drop a note of that in the Pearl Dive, and we try to bring extra attention to the question. Alas, it is too early to tell, whether the Pearl Dive will succeed.

Another entrant to the Pearl Dive. I thought I knew everything there is to know about finite abelian $p$-groups, but …. :-) Scratching this. While the question generated some interest, it seems evident to me that the asker is playing games with us, and the question was not genuine.

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Here's another curiosity "hijacked" into the Pearl Dive. When can we find sequences modulo a prime $p$ that obey the Fibonacci recurrence, and additionally have no repeated entries within the period?

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Yet another number theoretic curiosity. Are the numbers $\Phi_n(n)$ square-free for all $n$? Here $\Phi_n(x)$ is the $n$th cyclotomic polynomial.

Answer 10

While it hasn't received a ton of attention over at Math.SE, I'm particularly fond of my question about whether congruent rational triangles have a mutual rational dissection, since it seems so tantalizingly within reach — these triangles are constrained just enough that it feels like a proof 'should' be out there. (Even a proof of impossibility in the quadrilateral case, along the lines of the example I offered, seems like it would be a major victory.) The problem is unabashedly 'recreational' mathematics, but it's the sort that seems to touch very close to some quite serious math indeed (e.g., the parametrizations by rational points on elliptic curves in http://nyjm.albany.edu/j/1998/4-1p.pdf). I'd love to get more eyes on it if anyone has some thoughts.

Answer 11

This is a question that I had posted on MathSE. Other than the first comment I got no other response. I tried the suggestions in the comment but could go no further than a certain point. I updated the question accordingly and offered a bounty but still did not receive an answer.I hope this thread brings attention to the post. Despite it not having many upvotes I think it is interesting and should be up here, but if anyone can give me a good reason otherwise I shall take it down. Here it is -

Does Nagata theorem hold in a field that is not algebraically closed

Nagata's theorem states that if $R$ is a finitely generated $k$ algebra and $G$ is a geometrically reductive group acting rationally on $R$ then $R^G$ is finitely generated.

The question is if it holds when $k=\mathbb R$.

Thank you.

Answer 12

This question has remained unanswered, in-spite of a bounty.

The question defines homogenous polynomials on directed graphs and seeks to obtain references for literature on such polynomials.

I was considering posting on Mathoverflow, but this link inspired me to first post it here and get some attention.

Also, it would be nice to know if you (the reader) consider it fit for MO.

Answer 13

Is there a direct proof that a compact unit ball implies automatic continuity?

We know that if a Banach space is infinitely dimensional, then there are discontinuous linear functionals on it. Therefore having only continuous functionals imply finite dimension (at least in the presence of choice).

At the same time, we know that a compact unit ball must imply finite dimension, and therefore every linear functional is continuous.

Is there a direct proof from the compactness of the unit ball, that every linear functional is continuous?

Answer 14

I posted this question on math.SE and it has not been answered. I thought I should move it to MO. But I read this and I decided to follow Asaf Karagila's suggestion, so I am posting my question here.

There is a theorem by Poincaré and Hurwitz which says, basically, that an elliptic $\mathbb{Q}$-curve with positive rank has infinitely many points in every open subset of $\mathbb{P}^2\left (\mathbb{R}\right )$. I have not been able to find a proof for it which is written in English. I am looking for some reference.

Answer 15

My Q. about a problem with the proof of the lemma in subsection 3.3. of chapter 2, section 3 on sets of divergence in Katznelson's Introduction to Harmonic Analysis was looked at by nearly 100 people after I set a bounty, but except for an up-vote of the question there was until now no reaction (no answer, no comment, the only comment is mine).

The lemma is a IFF statement. The first part of the proof I found OK. The second part going in the other direction seems to me to have a subtle flaw, which may-be no one else has discovered yet (or those who have didn't communicate about it). A lot of often mentioned consequences in the rest of the section depend on the lemma, needing apparently both sides of the equivalence.

The difficulty may be hidden to many readers by the use of Landau's small & large O-notations in a remark made immediately before section 3.3. These notations don't let clearly mention how is the behaviour with respect to a parameter (not the variable involved in the behaviour described by the notation). The author seems to assume in his proof of the lemma (2nd part) a kind of uniformity with respect to the parameter which is an arbitrary point of the set E considered, although this uniformity is not asserted (at least not clearly) in the remark … and I don't see how it could be justified (while the remark itself is OK if not considered as including an unstated uniformity).

Nobody seeming to know the answer to my question, I wonder if the proof is in fact incomplete and may-be even the lemma and what follows might be false, which would have remained undetected because the result isn't the kind of things that has practical applications. (On the other hand, Katznelson was apparently not the only one working on the subject, so there might be other proofs that are OK.) In case how could I ask the author — born 1934, still alive by default, according to Wikipedia — an earlier email he didn't answer (may-be not enough important) … could someone do this for me with a better chance to get an answer from Y.K.?

Link to my orig. Q.: Difficulty in proof of a lemma in Katznelson's book about Harmonic Analysis chapt. 2 section 3 (divergence sets).

Answer 16

Solving Special Function Equations Using Lie Symmetries

This question is asking for a unified, intuitive, & visual, exposition of the classical special function equations (e.g. Hypergeometric) taking ideas from Lie theory, along the lines of the example of Bessel's equation as provided in the question. Lots of interest, no answers, awesome idea! Thoughts in the comments welcome.

Answer 17

I followed the link to this thread in order to post a question Which is Interesting, but not sufficiently answered. Here is the question. I want to know intuitively why does dividing the groups into two again and again gives minimum attempts to always get the correct switch. Also, what does it mean for minimum attempts here? Also I asked if I can post the question on Math Overflow again here.

Answer 18

Given two fixed boolean algebras $\mathfrak{A}$, $\mathfrak{B}$ I define a poset, consisting of pairs $(\alpha;\beta)$ of functions $\alpha:\mathfrak{A}\rightarrow\mathfrak{B}$, $\beta:\mathfrak{B}\rightarrow\mathfrak{A}$ such that (for every $X\in\mathfrak{A}$, $Y\in\mathfrak{B}$) $$Y\sqcap^{\mathfrak{B}}\alpha(X)\ne\bot^{\mathfrak{B}} \Leftrightarrow X\sqcap^{\mathfrak{A}}\beta(Y)\ne\bot^{\mathfrak{A}}.$$ Is this poset also a boolean lattice?

If this conjecture does not hold in general, does it hold for: a. atomic boolean lattices? b. atomistic boolean lattices? c. complete boolean lattices?

For the special case when $\mathfrak{A}$, $\mathfrak{B}$ are complete atomic boolean lattices, the conjecture follows from this answer.