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Classifications of cubic surfaces

Tue, 11/21/2017 - 06:46

Is there a known classification of singular cubic surfaces over finite fields?

Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

Tue, 11/21/2017 - 06:25

This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive answer (if not too complicated) would give a very efficient primality test (see the linked question for details).

Recall that the Chebyshev polynomials $T_n(x)$ are defined by $T_0(x)=1$, $T_1(x)=x$ and $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$, and there are several explicit expressions for their coefficients. Rather than writing them down (you can find them at the Wikipedia link anyway), let me just give a couple of examples: $$ T_{15}(x)=-15x(1-4\frac{7\cdot8}{2\cdot3}x^2+4^2\frac{6\cdot7\cdot8\cdot9}{2\cdot3\cdot4\cdot5}x^4-4^3\frac{8\cdot9\cdot10}{2\cdot3\cdot4}x^6+4^4\frac{10\cdot11}{2\cdot3}x^8-4^5\frac{12}{2}x^{10}+4^6x^{12})+4^7x^{15} $$ $$ T_{17}(x)=17x(1-4\frac{8\cdot9}{2\cdot3}x^2+4^2\frac{7\cdot8\cdot9\cdot10}{2\cdot3\cdot4\cdot5}x^4-4^3\frac{8\cdot9\cdot10\cdot11}{2\cdot3\cdot4\cdot5}x^6+4^4\frac{10\cdot11\cdot12}{2\cdot3\cdot4}x^8-4^5\frac{12\cdot13}{2\cdot3}x^{10}+4^6\frac{14}{2}x^{12}-4^7x^{14})+4^8x^{17} $$ It seems that $n$ is a prime if and only if all the ratios in the parentheses are integers; this is most likely well known and easy to show.

The algorithm described in the above question requires determining whether, for an odd $n$, coefficients of the remainder from dividing $T_n(x)-x^n$ by $x^r-1$, for some fairly small prime $r$ (roughly $\sim\log n$) are all divisible by $n$. In other words, denoting by $a_j$, $j=0,1,2,...$ the coefficients of $T_n(x)-x^n$, we have to find out whether the sum $s_j:=a_j+a_{j+r}+a_{j+2r}+...$ is divisible by $n$ for each $j=0,1,...,r-1$.

The question then is: given $r$ and $n$ as above ($n$ odd, $r$ a prime much smaller than $n$), is there an efficient method to find these sums $s_j$ without calculating all $a_j$? I. e., can one compute $T_n(x)$ modulo $x^r-1$ (i. e. in a ring where $x^r=1$) essentially easier than first computing the whole $T_n(x)$ and then dividing by $x^r-1$ in the ring of polynomials?

(As already said, only the question of divisibility of the result by $n$ is required; also $r$ is explicitly given (it is the smallest prime with $n$ not $\pm1$ modulo $r$). This might be easier to answer than computing the whole polynomials mod $x^r-1$.)

Solutions to a certain Birkhoff-interpolation problem

Tue, 11/21/2017 - 05:58

$\newcommand{\CC}{\mathbb{C}}$

Let for $n > 1$ and $m = n-1$ $$ p = x^n + a_1 x^{n-1} + \cdots + a_m x $$ be a polynomial with $a_i \in \CC$. Call $p^{(i)}(x) = \frac{d^ip}{dx^i}(x)$.

The following question would have an impact on the problem, if sequences of resultants $\mathrm{res}_x(p(x), p^{(n-i)}(x))$, $i = 1,\ldots,s$ are regular sequences:

Take $s < m$ and consider the space $X$ of monic polynomials $p$ as above such that $\alpha_1,\ldots,\alpha_s \in \CC$ (not necessarily distinct, could be even all equal) exist so that

\begin{equation} p^{(n-1)}(\alpha_1) = p^{(n-2)}(\alpha_2) = \cdots = p^{(n-s)}(\alpha_s) = 0 \end{equation}

and \begin{equation} p(\alpha_1) = \cdots = p(\alpha_s) = 0 \end{equation}

Question:

Can one show that for a certain solution $p(x) \in X$ with $a_{s+1} = 0$ there exists in every neighbourhood (identifying $X \cong \CC^m$) a solution in $X$ with $a_{s+1} \neq 0$?

This is obviously the case for $2 s + 1 \leqslant m$ where one can think of the $\alpha_i$ as fixed and consider the problem as a statement of linear algebra. If $2 s + 1 > m$ it may become necessary to "move the $\alpha_i$ a bit", to get $a_{s+1} \neq 0$.

I did some computer calculations with Maple and it always seemed to be possible to do such a small move, leading to $a_{s+1} \neq 0$, but I could not find a general proof.

Is the answer to the above question maybe already known to specialists in (Birkhoff-)interpolation? What could be helpful to read for making progress on this problem?

Extracting a divergent subsequence [on hold]

Tue, 11/21/2017 - 05:41

Let $(x_n)$ be a real sequence in $l^p$ for a fixed $p>1$ but not in $l^1$.

Is there a way to extract a sequence $(x_{\phi(n)})$ which is $l^{1+\varepsilon}$ and not $l^{1-\eta}$? (or a counter example)

Do prime ideals in polynomial ring generate prime ideals in the ring of holomorphic functions?

Tue, 11/21/2017 - 05:35

Suppose that $I \subset \mathbb C[z_1,\dots, z_n]$ is a prime ideal. Consider the ideal $I_{hol}$ in the ring of holomorphic functions $f: \mathbb C^n\to \mathbb C$ generated by polynomials from $I$.

Is $I_{hol}$ prime?

Sharp constant for inequality with convex functions

Tue, 11/21/2017 - 05:04

This is a follow up to this question, where the optimal constant was left open.

Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let \begin{equation} \mathcal{H} := \{f : P \rightarrow \mathbb{R} : f\text{ is convex and }\int_P f d\lambda = 0\} \end{equation} Question:

What is the largest possible constant $\alpha > 0$ purely depending on $n$ and $P$ such that \begin{equation} \forall f \in \mathcal{H}: \int_P \left| f \right| d\lambda \geq -\alpha \inf_{P} f \end{equation}

Notes:

The question is the same if one restricts $\mathcal{H}$ to functions with $\inf_P f = -1$.

This answer provides a lower bound of $\alpha \geq 4^{-n-1} |P|$. @fedja already mentions that a similar argument can be used to obtain a sharp constant, but I haven't been able to work it out. Hence this question.

Degree of vertex boundary on Hamming Cube

Tue, 11/21/2017 - 04:37

Let's take the Hamming graph $Q_n=(V,E)$, and take some subset $S \subseteq V$. Now, an inequality of Bobkov says that the vertex boundary of $S$ (number of neighbours of vertices in $S$) and the edge boundary of $S$ (number of edges from vertices of $S$ to vertices outside $S$) can't both be too small. I was wondering if we can include somewhere there the number of edges between the vertices of the vertex boundary. For instance, for a given size $2^k$, we know that among all sets of such size $k$-dimensional subcubes are the ones that minimize the edge boundary. For such a subcube, though, not only is the vertex boundary large, $(n-k)|S|$, but there are many edges between the vertices of the vertex boundary.

On the other hand, we can take a ball of radius $r$, which has a small edge boundary and large vertex boundary, and the vertex boundary forms an independent set.

So I guess my question is: can we say that for some subset $S$ of a given size, the vertex boundary and the number of edges between vertices of the vertex boundary (maybe normalized somehow) can not both be too small? what if we assume that $S$ is connected? monotone?

A good reference to the general Chinese Remainder Theorem

Tue, 11/21/2017 - 03:55

I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following

General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive integers $b_1,\dots,b_n$ the intersection $\bigcap_{i=1}^n(a_i+b_i\mathbb Z)$ is not empty if and only if $a_i-a_j\in d_{i,j}\mathbb Z$ for any $1\le i<j\le n$, where $d_{ij}$ denotes the largest common divisor of the numbers $b_i,b_j$.

Looking at the internet, I found this paper in which the General Chinese Remainder Theorem is formulated as an exercise and another paper in which this theorem is proved.

But I am suspecting that such General Chinese Remainder Theorem should be proved in some standard (undergraduate) textbook in Number Theory. I need it for a proper reference. Please help!

I understand that this is rather a reference request and not a problem of research level. In case of downvotes I will delete it as soon as will get a proper answer from experts.

$q$-analog of an integral from quantum field theory?

Tue, 11/21/2017 - 03:48

Let the function $f_q(x,y,z)$ completely symmetric in $x,y,z$ depend on $e^x,e^y,e^z$ only ($q$ is considered as a parameter). It is known that $f_q$ is a finite combination of elementary functions and theta functions

$$\theta_q(z)=(q;q)_\infty(z;q)_\infty(q/z;q)_\infty$$

and has the following properties $$ \begin{align} &f_q(x+c,y+c,z+c)=e^cf_q(x,y,z),\tag{1}\\ \\ &f_q\left(\frac{2\pi i}{3},0,z\right)=\left(e^z-1\right) \frac{\theta_q\left(e^{\frac{4 i \pi }{3}-z}\right)}{\theta_q\left(e^{-z}\right)},\tag{2}\\ \\ &f_q(x,y,z)=e^x+e^y+e^z\\ &+\frac32\left(e^x+e^y+e^z\right)\left(1+e^{x-y}+e^{y-x}+e^{y-z}+e^{z-y}+e^{z-x}+e^{x-z}\right)q\\ &+i\frac{\sqrt{3}}{2}\left(e^{-x}+e^{-y}+e^{-z}\right)\left(e^{2x}+e^{2y}+e^{2z}\right)q+O(q^2) \tag{3} \\ \end{align} $$

Q: Is it possible to find $f_q(x,y,z)$ from these data?

Note: According to comment by fedja the above conditions were not enough to determine the function $f$. Nevertheless is it possible to make further conjectures for the function $f$ from its definition in eq. $(5)$ or find it by calculating integral $(5)$ directly?

Motivation:

This question is related to $q$-analog of the following integral from quantum field theory due to F.A. Smirnov (also see this MSE post): $$ \begin{align} &\int_{-\infty}^{\infty}\prod_{j=1}^3\Gamma\Bigl(\frac 1 3 -\frac {\alpha-\beta_j}{2\pi i}\Bigr) \Gamma\Bigl(\frac 1 3 +\frac {\alpha-\beta_j}{2\pi i}\Bigr)(3\alpha-\sum\beta_m)e^{-\frac{\alpha}2}d\alpha\nonumber\\ &=\frac{(2\pi \Gamma(\frac 2 3))^2}{\Gamma(\frac 4 3)} \prod_{k\neq j}\Gamma\Bigl(\frac 2 3 -\frac {\beta_k-\beta_j}{2\pi i}\Bigr) e^{-\frac12\sum\beta_m}\sum e^{\beta_m},\quad |\text{Im}~\beta_j|<2\pi/3.\tag{4} \end{align} $$

Exact definition of $f_q$ is as follows

\begin{align} &\int_{-\infty}^\infty\prod_{j=1}^3\frac{\Gamma_q\left(\frac13+\frac{x-\beta_j}{2\pi i}\right)}{\Gamma_q\left(\frac23+\frac{x-\beta_j}{2\pi i}\right)}\frac{\sum_{m}q^{-\frac{\beta_m}{2 \pi i}}-\left(1+q^\frac13+q^{-\frac13}\right) q^{-\frac{x}{2 \pi i}}}{\prod_{m}\sin\left(\frac{\pi}3-\frac{x-\beta_m}{2 i}\right)}e^{-\frac{x}2}dx\\ &=\frac{-2\pi i}{(q;q)_{\infty }^9}\frac{\Gamma_q^2\left(\frac23\right)}{\Gamma_q\left(\frac43\right)}\frac{q^{5/9}}{(1-q)^2}e^{-\frac{1}{2} (\beta_1+\beta_2+\beta_3)} q^{-\frac{\beta_1+\beta_2+\beta_3}{6 \pi i}}\prod_{k\neq j}\Gamma_q\left(\frac23+\frac{\beta_j-\beta_k}{2\pi i}\right)\\ &\times\frac{\theta_q(q^{\frac{\beta_1-\beta_2}{2\pi i}})}{\sin\frac{\beta_1-\beta_2}{2 i}}\frac{\theta_q(q^{\frac{\beta_2-\beta_3}{2\pi i}})}{\sin\frac{\beta_2-\beta_3}{2 i}} \frac{\theta_q(q^{\frac{\beta_3-\beta_1}{2\pi i}})}{\sin\frac{\beta_3-\beta_1}{2 i}} \cdot f_{q_1}(\beta_1,\beta_2,\beta_3).\tag{5} \end{align} where $q_1=e^{-\frac{4 \pi ^2}{\ln(1/q)}}$ and $\Gamma_q$ is the $q$-Gamma function.

I have found strong evidence that $f_q$ has a relatively simple closed form expression.

Eq. $(2)$ is a condition that the difference of LHS and RHS of $(5)$ is an entire function. $(3)$ has been extracted from numerical calculations.

I could't find how to express $(5)$ in terms of q-series in a practical way (in principle it is possible to express $(5)$ in terms of $q$-series, but it contains a sum of $9$ $q$-hypergeometric functions ${}_3\phi_2$'s and $9$ of its derivatives plus $27$ Lambert series) and decided instead to guess the RHS and then prove it using Liouville's theorem.

Smooth proper fibration of complex projective varieties

Tue, 11/21/2017 - 03:35

Let $X$ be a smooth projective algebraic variety over the complex numbers.

(a) Do there exist:

a smooth proper map $\pi : \mathcal{X}\to S$ of algebraic varieties over the complex numbers, such that $\mathcal{X}$ and $S$ are smooth, $S$ is connected, $X$ is isomorphic to the fiber of $\pi$ over some $\mathbf{C}$-point of $S$, and such that there exists $s\in S(\mathbf{C})$, a $K$-scheme $X_0$, and an isomorphism of $\mathbf{C}$-schemes: $$(X_0)_{\mathbf{C}} \simeq \mathcal{X}_s$$

with $K$ either:

(a.1) $K = \mathbf{Q}$, (a.2) $K/\mathbf{Q}$ a finite extension, (a.3) $K = \bar{\mathbf{Q}}$?

(b) What can be reasonably called an "obstruction" to (a.i), $i = 1,2,3$?

Jacobian and configuration space and massey products

Tue, 11/21/2017 - 03:22

Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider $$ J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda. $$ Then $J^*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space $$ \operatorname{Conf}_{l}(J)\: : \: \operatorname{Conf}_{l}(X\setminus p) \to \operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda). $$ Let $V_{1}\subset H^{2}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})$ , resp. $V_{2}\subset H^{2}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a_{1}, \dots, a_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}_{l}(J)^{*}$ induces a surjection $$ \operatorname{Conf}_{l}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})\oplus V_{1}? $$ Notice that $\mathbb{C}$ represents the $0$-th cohomology group.

Pushforward maps for cohomology of coherent sheaves

Tue, 11/21/2017 - 03:01

Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map.

For any locally free coherent sheaf $\mathcal{F}$ on $X$, there is a pullback map $$\imath^*: H^i(X, \mathcal{F}) \to H^i(Z, \iota^* \mathcal{F});$$ and via Serre duality we have isomorphisms $H^i(X, \mathcal{F})^\vee = H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)$ and $H^i(Z, \iota^* \mathcal{F})^\vee = H^{m-i}(Z, \iota^* \mathcal{F}^\vee \otimes \omega_Z)$, where $\omega_X$ and $\omega_Z$ are the dualising sheaves. Setting $j=m-i$ and $\mathcal{G} = \mathcal{F}^\vee$, we conclude that there is a pushforward map $$\imath_*: H^j(Z, \iota^* \mathcal{G} \otimes \omega_Z) \to H^{j + c}(X, \mathcal{G} \otimes \omega_X),$$ for any $j$ and any locally free coherent sheaf $\mathcal{G}$ on $X$, where $c = n-m$ is the codimension of $Z$ in $X$.

Does this map have an intrinsic description (not using Serre duality)? Can it be defined without assuming that $X$ be projective, or that $\mathcal{G}$ be locally free?

Reference request for a binomial identity

Tue, 11/21/2017 - 02:02

I stumbled upon the following (perhaps well-known) identity for a positive integer $k$: $$\sum_{j=0}^n\frac{1}{(k-1)j+1}\binom{kj}{j}\binom{k(n-j)}{n-j}=\frac{1+kn}{1+(k-1)n}\binom{kn}{n}.$$

Could you please give me references where I can find a proof?

Could we solve the vector value ODE in a approximation way?

Tue, 11/21/2017 - 01:57

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

Am I allowed to say "first-order Vopěnka cardinal"?

Tue, 11/21/2017 - 01:11

For a cardinal $\kappa$ such that $V_{\kappa}$ satisfies Vopěnka's principle as a first-order axiom schema, am I allowed to say "first-order Vopěnka cardinal", or is there any kind of standard term for it?

Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?

Mon, 11/20/2017 - 22:21

It is well known that there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$, and that this embedding cannot be $C^2$.

Is there a $W^{2,2}$ isometric embedding? (i.e an isometric map $f \in W^{2,2}(\mathbb{T}^2,\mathbb{R}^3)$?)

Countably infinite connected Hausdorff space with the fixed point property

Sun, 11/19/2017 - 12:43

Is there an infinite, countable connected $T_2$-space $(X,\tau)$ such that $(X,\tau)$ has the fixed point property? (This means that for every continuous map $f:X\to X$ there is $x\in X$ such that $f(x) = x$.)

Fraction of the sets receive each color

Sun, 11/19/2017 - 02:57

Let $S_1,S_2,\dots,S_k$ be subsets of the set $S=\{1,2,\dots,n\}$, not necessarily distinct. We will color each element of $S$ red, green, or blue. From this coloring, each set $S_i$ will receive one or more color according to the following rule:

Let $r_i,g_i,b_i$ denote the number of red, green, and blue elements of $S_i$, respectively, and let $m_i=\max(r_i,g_i,b_i)$. If $r_i\geq m_i-1$, we give the color red to $S_i$. Similarly for green and blue.

What is the maximum constant $d$ for which we can always color the elements of $S$ in such a way that for any color, at least a fraction $d$ of the sets $S_i$ receive that color?

An algorithm that starts with a two-coloring and change the color of one element at a time to the third color achieves $d=1/5$, while an example shows that $d=1/3$ is the best one can hope for.

Band Limited, frequency phase zero-correlated, random process

Sun, 11/19/2017 - 00:06

I recently heard the following statement in a talk:

A complex random process is stationary if its Fourier transform is band limited and the phases are delta correlated. In that case, the correlation length of the process is proportional to the bandwith of the Fourier transform.

Where, as I understand it:

  • phase delta correlated means that if the process Fourier transform is $\hat{\psi} (\omega) = r_{\omega}e^{i\varphi _{\omega}}$, then $$\left(\varphi_{\omega}, \varphi_{\nu}\right)_{L^2} = \delta (\omega -\nu) \, .$$

  • The $L^2$ product is over the probability space (i.e., all possible realizations).

  • Stationary means that its correlation function $R(t,t')$ obeys $R(t,t') = r(|t-t'|)$.

Because it was stated very informally, and I have almost no background in stationary processes, I have three, basic questions:

  1. Is this true as stated, or does it need further refinements to be exact? Could you refer to a proof?
  2. Is this condition also necessary, or only sufficient?
  3. What assures us that the Fourier transform exists for every realization?

A more dense analog of the Mills' constant

Sat, 11/18/2017 - 07:47

Is there a real number $A$ such that $$\left \lfloor n^{A} \right \rfloor$$ is a prime number ($\forall n \in \mathbb{N})$? It is obvious that $A>1+\epsilon$ from the prime number theorem.

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