# Linked Questions

47 questions linked to/from Proposals for polymath projects
517 votes
3 answers
52k views

### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
• 5,213
63 votes
3 answers
4k views

### Are there infinitely many integer-valued polynomials dominated by $1.9^n$ on all of $\mathbb{N}$?

The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of ...
• 13.5k
51 votes
6 answers
6k views

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ... • 887 70 votes 2 answers 6k views ### Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used? I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ... • 25.6k 37 votes 3 answers 8k views ### The error in Petrovski and Landis' proof of the 16th Hilbert problem What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post.. For Mathematical ... • 119 21 votes 6 answers 2k views ### Consistency strength needed for applied mathematics Given that we can never proof the consistency of a theory for the foundations of mathematics in a weaker system, one could seriously doubt whether any of the commonly used foundational frameworks (ZFC ... • 1,021 10 votes 5 answers 4k views ### Applications of functional analysis beyond analysis(towards algebra, geometry, number theory...) [closed] So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional ... 57 votes 1 answer 5k views ### Can you solve the listed smallest open Diophantine equations? In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ... • 3,155 12 votes 2 answers 1k views ### Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic) [closed] This is a question about the proofs of Kazhdan-Lusztig's conjectures for category$\mathcal{O}$using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory). ... • 1,756 18 votes 3 answers 5k views ### Number of unique determinants for an NxN (0,1)-matrix I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ... • 359 11 votes 2 answers 1k views ### Elliptic operators corresponds to non vanishing vector fields Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ... • 119 17 votes 1 answer 924 views ### Distinct integer roots for a degree 7+ polynomial and its derivative Question: Is there a polynomial$f \in \mathbb{Z}[x]$with$\deg(f) \geq 7$such that all roots of$f$are distinct integers; and all roots of$f'$are distinct integers? Background: I asked a ... • 7,296 13 votes 2 answers 1k views ### Average degree of contact graph for balls in a box Imagine you dump congruent, hard, frictionless balls in a box, letting gravity compress the balls into a stable configuration (I believe such configurations are called jammed.) Assume the box ... • 146k 26 votes 2 answers 2k views ### Codimension of the range of certain linear operators Assume that$P(x,y), Q(x,y) \in \mathbb{R}[x,y]$are two polynomials. We define a linear map$D$on$\mathbb{R}[x,y]$with$D(U)=PU_{x}+QU_{y}$. In fact$D$is the differential operator corresponding ... • 119 21 votes 1 answer 515 views ### Existence of a polynomial$Q$of degree$\geq (p-1)/4$in$\mathbb F_p[x]$such that$QQ'$factorizes into distinct linear factors For all primes up to$p=89$there exists a product$Q=\prod_{j=1}^d(x-a_j)$involving$d\geq (p-1)/4$distinct linear factors$x-a_j$in$\mathbb F_p[x]$such that$Q'$has all its roots in$\mathbb ...

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