Recent MathOverflow Questions

Order of Quantifiers in Axioms of a Vector Space

Math Overflow Recent Questions - Thu, 01/11/2018 - 18:50

Conventionally, a vector space over a field $F$ is a set $V$ together with two operations $+:V\times V\to V$ and $\cdot:F\times V\to V$ satisfying

  1. $(a+b)+c = a+(b+c)$ for all $a,b,c\in V$
  2. $a+b=b+a$ for all $a,b\in V$
  3. There exists an element $\vec 0\in V$ so that $a+\vec 0=a$ for all $a\in V$
  4. For all $a\in V$ there exists $-a\in V$ satisfying $a+(-a) = \vec 0$ (with $\vec 0$ satisfying axiom 3)
  5. $\alpha\cdot (a+b)=\alpha\cdot a+\alpha\cdot b$ for all $a,b\in V$ and $\alpha\in F$
  6. $(\alpha+\beta)\cdot a=\alpha\cdot a+\beta\cdot a$ for all $a\in V$ and $\alpha,\beta\in F$
  7. $(\alpha\beta)\cdot a=\alpha\cdot (\beta \cdot a)$ for all $a\in V$ and $\alpha,\beta\in F$
  8. $1\cdot a=a$ for all $a\in V$.

Then we prove that $\vec 0$ is unique, which justifies the sloppiness in stating axiom 4.

I recently made a careless quantifier error, defining the following:

A Q-space over a field $F$ is a set $V$ together with two operations $+:V\times V\to V$ and $\cdot:F\times V\to V$ satisfying for all $a,b,c\in V$ and $\alpha,\beta \in F$

  1. $(a+b)+c = a+(b+c)$
  2. $a+b=b+a$
  3. There exists an element $\vec 0_a\in V$ so that $a+\vec 0_a=a$
  4. There exists $-a\in V$ satisfying $a+(-a) = \vec 0_a$ (with $\vec 0_a$ satisfying axiom 3)
  5. $\alpha\cdot (a+b)=\alpha\cdot a+\alpha\cdot b$
  6. $(\alpha+\beta)\cdot a=\alpha\cdot a+\beta\cdot a$
  7. $(\alpha\beta)\cdot a=\alpha\cdot (\beta \cdot a)$
  8. $1\cdot a=a$

I suspect that a Q-space is different than a vector space. My question is, can anyone give an example of a Q-space that isn't a vector space? I do know that if axiom 4 of a Q-space is changed to

  1. There exists $-a\in V$ so that $b+a+(-a)=b$

we can then prove $\vec 0_a=\vec 0_b$, making the modified Q-space the same as a vector space.

Function For Data Points

Math Overflow Recent Questions - Thu, 01/11/2018 - 18:14

I would like to define a function for the following data points: {0,0},{1,200},{2,266},{3,300},{7,350}

As you can see as X increases, Y also increases but at a smaller and smaller amount as X grows. I would like to be able to calculate Y at any point along this curve.

Wolfram Alfa tries to create a curve but fails.

Is it consistent with ZF that $V\to V^{\ast \ast}$ is always surjective?

Math Overflow Recent Questions - Thu, 01/11/2018 - 16:15

In a comment to a recent question, Jeremy Rickard asked whether it is consistent with ZF that the map $V \to V^{**}$ from a vector space to its double dual is always surjective. We know that "always injective" is consistent (since that's what happens in ZFC) and Jeremy Rickard's argument shows that "always an isomorphism" is not consistent. But what about "always surjective"?

How to measure distribution of high-dimensional data

Math Overflow Recent Questions - Thu, 01/11/2018 - 16:12

I have to methods of projecting random samples in $\mathbb{R}^n$ onto a manifold defined by $C(q)=0$, which is a lower-dimensional subset. Now, samples in $\mathbb{R}^n$ are uniformly distributed. However, when projecting on to the manifold, the distribution is not guaranteed to be uniform. The two projection methods will provide different distributions.

My question is, what is a good measure of distribution that will enable comparison between the two methods? In simple words, I have two high-dimensional data-sets, how can I compare their distributions?

Good books on the divisor sum function $\sigma(n)$?

Math Overflow Recent Questions - Thu, 01/11/2018 - 15:43

I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that was made in the past centuries.

I am interested in arithmetic functions in general so the books mentioned in this question Good books on Arithmetic Functions ? is already a good start.

However, it would be great if you could recommend a book or a comprehensive paper that introduces/summarizes knowledge about the $\sigma(n)$ function.

Who first used the words "Endomorphism" and "Automorphism"?

Math Overflow Recent Questions - Thu, 01/11/2018 - 15:37

Who first used the words "endomorphism" and automorphism" as a specialization of "homomorphism" ?

equidistribution of the number of occurrences of a vincular pattern, and a simpler vincular pattern

Math Overflow Recent Questions - Thu, 01/11/2018 - 15:00

This is (at least for now) a question out of curiosity, there is no "deeper" meaning to it I know of. In fact, my main question is: is the observation below obvious?

To state the observation I have to define two statistics on permutations $|1|23:\mathfrak S_n\to \mathbb N$ and $|123:\mathfrak S_n\to \mathbb N$, and two maps, $K:\mathfrak S_n\to\mathfrak S_n$ and $S:\mathfrak S_n\to\mathfrak S_n$.

Let $\pi$ be a permutation, then an occurrence of the vincular pattern $|1|23$ (warning: notations vary) is an occurrence of the ordinary pattern $123$ such that the first matched entries are the first two entries of the permutation. In other words the number of occurrences of $|1|23$ in $\pi$ is zero, if the $\pi(2) < \pi(1)$, and it is the number of entries larger than $\pi(2)$ otherwise. The statistic https://findstat.org/St001084 counts the number of occurrences of $|1|23$ in $\pi$.

Similarly, an occurrence of the vincular pattern $|123$ is an occurrence of the ordinary pattern $123$ such that the first matched entry is the first entry of the permutation. The statistic https://findstat.org/St000804 counts the number of occurrences of $|123$ in $\pi$.

Now, for the maps! Let $K$ be the inverse Kreweras complement http://findstat.org/Mp00089 mapping $\pi$ to $(1,\dots,n)\pi^{-1}$, and let $S$ be the Simion-Schmitt http://findstat.org/Mp00068 map, sending any permutation to a $123$ avoiding permutation.

Observation:

At least for $n\leq 8$, the distribution over $\mathfrak S_n$ of the number of occurrences of $|1|23$ is the same as the distribution of $|123\circ K\circ S$.

Why would this be the case? A bijective argument might be especially nice!

To what extent does Floer cohomology detect Hamiltonian non-displaceability of immersed curves?

Math Overflow Recent Questions - Thu, 01/11/2018 - 14:56

Floer cohomology for immersed Lagrangians is introduced by Akahi, Manabu; Joyce, Dominic, Immersed Lagrangian Floer theory, J. Differ. Geom. 86, No. 3, 381-500 (2010) and its one-dimension version (in the absence of "teardrops") is developed in Abouzaid, Mohammed, On the Fukaya categories of higher genus surfaces, Adv. Math. 217, No. 3, 1192-1235 (2008). See also the embedded case described in de Silva, Vin; Robbin, Joel W.; Salamon, Dietmar A., Combinatorial Floer homology. As in the embedded case, non-triviality of Floer cohomology of immersed Lagrangians obstructs Hamiltonian displaceability. For example, consider the following example of an immersion $f: L \to X $ in the two sphere $X$ (thought of as the one-point compactification the plane) dividing the two sphere into areas $A_0,A_1,A_2,A_3$ (smoothing used curve-shortening by A. Carapetis)

an immersed curve

A computation shows that the Floer cohomology admits a weakly bounding cochain and is Floer-non-trivial if $A_2 < \min(A_1,A_3)$ and

$$A_0 = A_1 + A_3 - 2A_2.$$

Indeed in this case using the Morse model there are six generators in the Floer cochains, given by the max and min of the height function and four generators corresponding to the two self-intersection points; the zeroth Fukaya map is non-zero because of the two teardrops but the teardrops can be "killed" by taking a bounding cochain that is the sum of generators coming from the self-intersections with coefficients $q^{A_1 - A_2}$ and $q^{A_3 - A_2}$. In fact one can do a bit better, if I did the signs correctly, and get a family of such bounding cochains as long as

$$A_0 \in (A_1 + A_3 - 2A_2, A_1 + A_3 - A_2) .$$

(Take the bounding cochain to be such that the region with area $A_0$ contributes to the deformed zeroth Fukaya composition map.) On the other hand, Moser's principle Moser, Jürgen, On the volume elements on a manifold, Trans. Am. Math. Soc. 120, 286-294 (1965) implies that $f(L)$ is non-displaceable iff each of the areas is at most the sum of the other three:

$$ A_i \leq A_j + A_k + A_l, \quad i,j,k,l \ \text{distinct}. $$

Indeed, if one of these inequalities fails then there is a symplectomorphism that takes the complement of the interior of one of these regions into its interior, while if they all hold then such a symplectomorphism obviously does not exist. The conditions

$$A_0 + 2A_2 = A_1 + A_3, A_2 < A_1, A_2 < A_3$$

imply the "Moser conditions" (exercise), but are much stronger. So there seems to be a fairly big gap between Floer non-triviality and non-displaceability in this case.

Question: (1) Can anyone do better, i.e. is it possible to detect the non-displaceability of immersed curves in the two-sphere using some other version of Floer theory, or a different weakly bounding cochain?

(2) Suppose one takes the product of two such curves in $S^2 \times S^2$. Here Floer theory still works, but there is no analog of Moser's results although presumably one could use McDuff, Dusa, Displacing Lagrangian toric fibers via probes, Proceedings of Symposia in Pure Mathematics 82, 131-160 (2011). When is the image of an immersion of product form displaceable?

(3) Is there a description of the space of Maurer-Cartan solutions for an immersed curve in the two-sphere, or more generally a Riemann surface? For example, (is there a good notion of dimension of the space of Maurer-Cartan solutions and if so) what is the dimension?

Pullback of invariant differential under addition of isogenies

Math Overflow Recent Questions - Thu, 01/11/2018 - 14:28

Let $H$ and $G$ be two commutative group varieties over a field $K$, and let $\psi, \phi$ be two morphisms of group varieties $\psi, \phi: H \rightarrow G$. Let $\psi + \phi: H \rightarrow G$ be the addition map: $$ H \xrightarrow{(\psi,\phi)} G \times G \xrightarrow{+} G. $$ Let $\omega$ be a nonzero regular differential form on $G$.

My question is the following:

  1. Is there a relation between $(\psi + \phi)^*\omega$ and $\psi^* \omega + \phi^* \omega$? are they always equal?
  2. If this is not true in general, then why does it hold in the case when $H = E_1, G = E_2$ are two ellitpic curves?
  3. To elaborate the situation of $2.$ above, I do not quite understand the proof given in Silverman's Arithmetic of Elliptic Curves Theorem III.5.2. To rephrase, I do not see how one comes up with the proof given there, or what is going on behind the proof, and in what generality this is true. It would be great if someone could give a short/conceptual proof.

I apologize in advance if this is not the correct level/type of questions to be asked here.

$c_0$-direct sum and projective tensor product

Math Overflow Recent Questions - Thu, 01/11/2018 - 13:53

Let $\cal I$ be an indexed set and $\cal{(A_i)_{i\in I}}$ be a collection of Banach algebras. Can we see the following topological isomorphism?

$$(c_0-\oplus_{i\in \cal I}A_i)\hat{\otimes}(c_0-\oplus_{i\in \cal I}A_i)\cong c_0-\oplus_{i\in \cal I}A_i\hat{\otimes}A_i.$$

If the answer is no, is there any conditions that this relation be true?

a continuity question concerning metrics on probablility measures

Math Overflow Recent Questions - Thu, 01/11/2018 - 13:24

For a metric space $M$, I'll write $Prob(M)$ for the Borel probability measures on $M$.
I am interested in metrics on $Prob(M)$, such as the Kantorovich distance (or other metrics).

If $f: M \rightarrow M'$ is continuous, is the induced map $Prob\ f : Prob(M) \rightarrow Prob(M')$ also continuous? Is it true when "continuous" is replaced by "uniformly continuous"?

Is this known for the Kantorovich metric? Or for any other metric?

I'm also interested case of discrete metrics. (This is the main application that I have in mind.)

Distribution of primitive roots, as p varies

Math Overflow Recent Questions - Thu, 01/11/2018 - 11:47

For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)

I am curious about the distribution of $\Phi(p)$ as $p$ varies. I imagine that this is classic analytic number theory, but I'm finding it surprisingly hard to locate a precise statement/reference.

To make my question precise, consider the following: For any continuous function $f \colon [0,1] \rightarrow {\mathbb R}$, define $$D_p(f) = \frac{1}{\phi(p-1)} \cdot \sum_{x \in \Phi(p)} f \left( \frac{x}{p-1} \right).$$

Does this sequence of distributions $D_p$ converge (e.g., weakly in measure, or in some stronger sense?) to the uniform measure or something else on $[0,1]$? What's the best current result along these lines?

How many non-homeomorphic collections of $N$ circles in $\mathbb{R}^3$ are there?

Math Overflow Recent Questions - Thu, 01/11/2018 - 07:31

Let's have a finite collection of $N$ circles $\mathbb{S}^1$ in $\mathbb{R}^3$. (These circles could not intersect.) Every circle could be "hooked on" other circle and it could be "hooked" for simplicity only once. My question, which I need to solve, is how many combinations of non-homeomorphic structures will I obtain? For example $N=2$: I have $2$ combinations, two unhooked circles and two hooked circles; I know already that this question could be translated to the language of graph theory. Do You know, if anyone has solved such a problem? Thank You!

Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

Math Overflow Recent Questions - Thu, 01/11/2018 - 07:29

Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not surjective. As we learned in this question, there are models of $ZF$ where $V \to V^{\ast \ast}$ is an isomorphism when $V$ has a countable basis. I think the same argument shows that it is consistent with ZF that this is an isomorphism whenever $V$ has a basis.

Is it consistent with $ZF$ that $V \to V^{\ast \ast}$ is an isomorphism for all vector spaces $V$?

I ask because I'm teaching a rigorous undergrad analysis class. My students keep asking me whether they have to believe that $V \to V^{\ast \ast}$ can fail to be an isomorphism. Of course, I'm trying to change their intuition to point out why most mathematicians find the failure of isomorphism plausible and point out that there are more subtle ways to salvage the claim, such as Hilbert spaces, but I'd also love to be able to give them a choice free proof that there is some vector space where this issue comes up.

Numerical evaluation/approximation of non-central high-order moments of high-dimensional Gaussian measures?

Math Overflow Recent Questions - Thu, 01/11/2018 - 04:15

I need to numerically evaluate/approximate non-central high-order moments of high-dimensional Gaussian measures/distributions with given mathematical expectations and covariance matrices. The Gaussian measures have dimension $d$ with say $d>1000$ and the moments have order $k$ with say $k>>1000$.

Hence the classical Wick-Isserlis theorem/formula looks useless because it would involve an astronomically large number of terms $k!!$.

So, please, what are my best options in order to numerically evaluate those moments in general?

My covariance matrices have a special stucture, they are pentadiagonal. Does it help in evaluating the moments?

Thanks.

Controlling subsolutions of a second order linear ODE

Math Overflow Recent Questions - Wed, 01/10/2018 - 14:54

Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality $$f'' - 2\alpha f' + 2\alpha f \leq 0$$ where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $f(x) < e^x - 1$ for some $x$?

Note that the solution to the corresponding differential equation oscillates since the characteristic equation has complex roots (call this solution $g$). Thus we can certainly say $g(x) < e^x -1$ for infinitely many $x$. My first thought was to try to control $f$ by $g$ a la Gronwall's inequality. However, I was recently shown that the analogue to Gronwall for degree two differential equations doesn't hold.

Any ideas would be welcome. Also, any good references for differential inequalities that might help me solve this problem are equally welcome.

Solution to algebraic equations over $\mathbb{C}$ and $\mathbb{C}[x]$

Math Overflow Recent Questions - Mon, 01/08/2018 - 18:14

$t^n=a$, we get one solution to the equation: $$t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$$, generalizing this result by replacing the exponential with an elliptic modular function and the integral with hyperelliptic integrals, we can get every solution to an algebraic equation $x$ in $a_0+a_1x+a_2x^2+\cdots+a_nx^n=0$ with degree above 5 by formulation of modular function and hyperelliptic integral(both formulated by Siegel Theta functions): $$y=\frac{\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}+\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}-\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}$$

or

$$y=\frac{1}{2}+\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}-\frac{\theta\left( \begin{array}{cccccc} 0 & 0 & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} 0 & \frac{1}{2} & \cdots & 0 \\ \frac{1}{2} & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}{2\theta\left( \begin{array}{cccccc} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4\theta\left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & \cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \end{array} \right)(\Omega)^4}$$ where $\Omega$ is the period matrix of the hyperelliptic curve $\mathbb{C}$ please see David Mumford Tate lecture on Theta $\textrm{II}$ Jacobian page 266 for more detail.

Now let us extend this result to algebraic equation with coefficients of $p(x) \in Q[x]$ where $Q[x]$ is a ring.

$$p_0(x) + p_1(x) \cdot y + p_2(x)\cdot y^2 + \cdots + p_n(x)\cdot y^n $$ is an algebraic polynomial where $p_i(x)$ are the polynomials with rational coefficients. When $$p_0(x) + p_1(x) \cdot y + p_2(x)\cdot y^2 + \cdots + p_n(x)\cdot y^n =0$$, we have solution to the equation in which $y$ is formulated by modular function and hyperelliptic integrals with $x$ as variable, like $y= \phi(x)$, in another word, $$p_0(x) + p_1(x) \cdot \phi(x) + p_2(x)\cdot \phi(x)^2 + \cdots + p_n(x)\cdot \phi(x)^n =0$$

My question is when $y$ is expanded as power series (Taylor expansion) in $x$,as $$y=\sum_0^{\infty }a_i x^i$$, or $$\phi(x) = \sum_0^{\infty }a_i x^i$$, under what condition ( formulated by modular function and hyperelliptic integrals ) can we have $a_i\in \mathbb{N}\bigcup 0$ ?

Bounding the dimension of the locus where a variety has larger than expected dimension

Math Overflow Recent Questions - Mon, 01/08/2018 - 18:10

Disclaimer: I am a research mathematician, but not an algebraic geometer, and so I don't know if this is a good question. I welcome advice for improving it and/or better tags.

Let $K$ be an infinite field (I'm imagining $K=\mathbb{C}$).

Let $P_1, \dots, P_n$ be polynomials in $K[t_1,\dots, t_k, x_1,\dots,x_m]$. In the examples I have in mind, these polynomials all have total degree $\leq 2$, so if assuming this makes this question easier to answer, feel free to do so.

If we specialize $(t_1,\dots,t_k)$ to a point $t$ in $\mathbb{A}^k$, then the system of equations $P_1=\dots=P_n = 0$ cuts out a variety $V_t$ in $\mathbb{A}^m$, which we would expect to be of dimension $\max\{0,m-n\}$. For $l = 1, 2, \dots,$ define $$S_l = \left\{t \in \mathbb{A}^k ~\big|~ \dim(V_t) \geq \max\{0,m-n+l\}\right\}.$$

  1. What kind of set is $S_l$? Is it algebraic? If so (or if not!), is there an algorithm to describe it?
  2. (Assuming this question makes sense) is there a general upper bound in terms of $l$ on the dimension of $S_l$?

Example: take $m=n$ and $k=n^2$, and for $i=1,\dots,n$ let $P_i = t_{i,1}x_1 + \dots + t_{i,n}x_n.$ Then $S_l$ is the locus where the matrix $(t_{ij})$ has rank $\leq n-l$, and this set is cut out by the vanishing of some determinants.

A question on heights and Northcott's Theorem

Math Overflow Recent Questions - Mon, 01/08/2018 - 17:09

Could anyone please point out some references and different proofs for Northcott's Theorem about finiteness of the number of points of bounded height in projective varieties over number fields, and generalizations (if any)?

I know some proofs, but I would like to inspect some references in light of the following question.

My main question, then, is: what are the properties of height functions, that make Northcott's Theorem work?

In other words, let $X$ be a projective variety over a number field $K/\mathbf{Q}$, and let $h : X(K)\to\mathbf{R}_{\ge 0}$ be a function.

Suppose $h$ satisfies the following property $\mathcal{P}$:

$$\#\{x\in X(K)\mid h(x)\le B\}$$ is a finite set for every $B\in\mathbf{R}_{\ge 0}$.

Can one find (necessary and) sufficient conditions on $h$ for $h$ to satisfy $\mathcal{P}$?

Example. If there is an ample line bundle $\mathscr{L}$ and $h = h_{\mathscr{L}}$ is the corresponding Weil height, then $h$ does satisfy $\mathcal{P}$ by Northcott's Theorem.

The idea is to extract, if possible, properties of height functions that make this implication still work. In other words, what are sufficient conditions (merely about $h$, ie. without assuming $h$ is a Weil height in the first place) to ensure $h$ has the finiteness property?

The question does not really ask for a solution to this task, but rather if this is already known to anybody.

Non-linear first order ODE $ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$

Math Overflow Recent Questions - Mon, 01/08/2018 - 17:01

I am trying to solve an ODE which has the following form: $$ \dfrac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$$ with an initial condition $y(x_0) = y_0 \\ $.

The approaches I could think of to solve this equation were to

  1. Approximate it to a Darboux equation, but the approximations are not desirable.
  2. Write $x = \dfrac{x}{z}$ and $y = \dfrac{y}{z} $ to get homogeneous degree on the right side, but I am unable to progress further.

Are there any methods to find explicit closed form solutions for such equations?

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