博客

Baragar-Bourgain-Gamburd-Sarnak 猜想：Markov triples 的模 p 连通性

旧博客原文

原题：Baragar-Bourgain-Gamburd-Sarnak conjecture

$M$ is the markov triple $(x,y,z)$ :

$x^2+y^2+z^2=xyz$ and $(x,y,x)\in \mathbb Z^3 \ \ \ \ (*)$ .

It is easy to see:

$R_1: (x,y,z)\to (3yz-x,y,z)$ .

map markov triple to markov triple.

This is also true for $R_2,R_3$ . and the transform $R_1,R_2,R_3$ and permutation a classical result of markov claim that all solution of (*) could be generated from $(1,1,1)$ . I get a similar result for a similar algebraic equation 1 half years ago when consider a $Q$ version of problem about 1-form given by Xu Bin.

Now we know the graph with root $(1,1,1)$ and with node generate by transform $R_1\cup R_2 \cup R_3 \cup S_3$ is connected.

The B-B-G-S conjecture is is the connected property still true for prime $p$ surfficed large?

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Markov triples 是方程

$$x^2+y^2+z^2=3xyz$$

的正整数解。经典 Markov 定理说，从根解 $(1,1,1)$ 出发，反复使用 Vieta involution

$$(x,y,z)\mapsto (x,y,3xy-z)$$

以及坐标置换，可以生成所有正整数解。

Markov triples 的 Vieta involution 生成解图；模 p 后的问题变成有限域曲面上的连通性问题。

1. 解图

把每个解看作图的一个顶点，若两个解由一次 Vieta involution 或置换相连，就连一条边。整数正解形成一棵以 $(1,1,1)$ 为根的巨大图。

2. 模 p 的问题

把方程放到有限域 $\mathbb F_p$ 上，得到有限集合

$$X_p=\{(x,y,z)\in\mathbb F_p^3:x^2+y^2+z^2=3xyz\}.$$

同样的 involution 仍然作用在 $X_p$ 上。Baragar-Bourgain-Gamburd-Sarnak 方向的问题是：当 $p$ 足够大时，这个作用图是否在主要部分上连通，甚至是否具有 expansion 性质？

3. 为什么这不是普通图论

图的边来自代数自同构，因此顶点集合有强烈的代数几何结构。连通性问题本质上是在问：这些简单变换能否在有限域曲面上产生足够大的轨道。

4. 可能工具

这类问题常混合使用代数几何、有限群展开、谱间隙和 sum-product 现象。若能证明没有大的不变子集，就能排除图分裂成多个大块的可能。

2017年11月22日
Metric entropy（二）：entropy map 的上半连续性与 infinity entropy
旧博客原文

原题：Metric entropy 2

I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”.

Now we focus on the upper semi-continuouty of the metric entropy map. The object we investigate is $(X,T,\mu)$ , where $\mu$ is a $T-$ invariant measure.

The insight to make us interested to this kind of problem is a part of variational problem, something about the existence of certain object which combine a certain moduli space to make some quantity attain critical value(maximum or minimum). The most simple example maybe Isoperimetric inequality and Dirichlet principle of Laplace. Any way, to establish such a existence result a classical approach is to proof the upper semi-continuouty and bounded for associate energy of the problem. In our case the semi-continuouty will be some thin about the regularity of the entropy map:

$E:M(X,T)\to h_{\mu}.$

We define the entropy at infinity:

$sup_{(\mu_n)}limsup_{\mu_n\to 0}h_{\mu_n}(T)$

Where $(u_n)_{n=1}^{\infty}$ varies in all sequences of measure coverage to $0$ in the sense for all $A\subset M$ , $A$ measurable then $\lim_{n\to \infty} \mu_{n}(A)=0$ .

Compact case

we say some thing about the compact case, In this case we have finite partition with smaller and smaller cubes, this could be understand as a sequences of smaller and smaller scales. A example to explain the differences is $\mathbb N^{\mathbb N},\sigma$ , shift map on countable alphabet.

Because of this thing, there is a good sympolotic model, i.e. h-expension, and it generalization asymptotically h-expension equipped on a compact metric space $X$ have been proved to be that the corresponding entropy map is upper semi-continous.

In particular $C^{\infty}$ diffeomorphisms on compact manifold is asymptotically h-expensive.

Natural problem but I do not understand very well:

Why it is natural to assume the measure to be probability measure in the non-compact space?

Non-compact case

$(X,d)$ metric space

$T:X\longrightarrow X$ is a continuous map.

$d_n(x,y)=\sup_{0\leq k\leq n-1}d(T^kx,T^ky)$ , then $d_{n}$ is still a metric.

Easy to see $\frac{1}{n}h_{\mu}(T^n)=h_{\mu}(T)$ . This identity could be proved by the cretition of entropy by $\delta$ -seperate set and $\delta$ -cover set.

Kapok theorem:

$X$ compact, for every ergodic measure $\mu$ the following formula hold:

$h_{\mu}(T)=\lim_{\epsilon \to 0}limsup_{n\to \infty}\frac{1}{n}logN_{\mu}(n,\epsilon,\delta)$ .

Where $h_{\mu}(T)$ is the measure theoretic entropy of $\mu$ .

Riquelme proved the same formula hold for Lipchitz maps on topological manifold.

Let $M_e(X,T)$ defined the moduli space of $T$ -invariant portability measure.

Let $M_(X,T)$ defined the moduli space of ergodic $T$ -invariant probability measure.

Simplified entropy formula:

$(X,d,T)$ satisfied simplified entropy formula if $\forall \epsilon >0$ surfaced small and $\forall \delta\in (0,1)$ , $\mu\in M _e(X,T)$ .

$h_{\mu}(T)=\limsup_{n\to \infty}\frac{1}{n}log(N_{\mu}(n,\epsilon,\delta))$ .

Simplified entropy inequality:

If $\epsilon>0$ suffciently small, $\mu \in M_{e}(X,T)$ , $\delta\in (0,1)$ .

$h_{\mu}(T)\leq \limsup_{n\to \infty}\frac{1}{n}log(N_{\mu}(n,\epsilon,\delta))$ .

Weak entropy dense:

$M_e(X,T)$ is weak entropy dense in $M(X,T)$ . $\forall \lambda>0$ , $\forall \mu\in M(X,T)$ , $\exists \mu_n\in M_e(X,T)$ , satisfied:
1. $\mu_n\to \mu$ weakly.
2. $h_{\mu_n}(T)>h_{\mu}(T)-\lambda$ , $\forall \lambda>0$ .
补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

entropy map 的上半连续性是变分问题中的关键正则性。若想证明某个 invariant measure 使 entropy 或 pressure 达到最大，通常需要紧性和上半连续性。

entropy map 的上半连续性关系到最大熵测度存在性；非紧情形还要控制 entropy at infinity。

1. Entropy map

给定动力系统 $f:X\to X$，考虑 invariant measures 空间 $\mathcal M_f(X)$ 上的函数

$$\mu\mapsto h_\mu(f).$$

若 $\mu_j\to\mu$ 弱收敛，希望有

$$\limsup_{j\to\infty}h_{\mu_j}(f)\le h_\mu(f).$$

这就是上半连续性。

2. 紧空间情形

在紧空间上，可以用越来越细的有限 partition 近似 entropy。若系统具有 expansiveness 或 asymptotic h-expansiveness，entropy map 常有较好的上半连续性。

3. 非紧空间的困难

非紧空间中，测度可能逃向无穷远。即使 $\mu_j$ 弱收敛到某个极限，entropy 也可能在逃逸部分携带额外信息。这个额外损失可用 entropy at infinity 衡量。

4. Entropy at infinity

粗略地说，entropy at infinity 记录所有逃向无穷远的测度序列可能保留的 entropy：

$$h_\infty=\sup_{\mu_j\to0}\limsup h_{\mu_j}(f).$$

若 $h_\infty$ 小于系统的 topological entropy，就有机会证明最大熵测度存在。

5. 几何动力系统中的意义

在 geodesic flow 中，entropy 与轨道增长、曲率和测地线逃逸相关。上半连续性问题本质上是在问：复杂轨道是否可能全部跑到 cusp 或无穷远处。若不能，就能在内部找到达到最大 entropy 的测度。
2017年11月21日
Metric entropy（一）：Ruelle 不等式、Pesin 公式与 Lyapunov 指数

旧博客原文

原题：Metric entropy 1

Some basic thing, include the definition of metric entropy is introduced in my early blog.

Among the other thing, there is something we need to focus on:

1.Definition of metric entropy, and more general, topological entropy.

2.Spanning set and separating set describe of entropy.

3.amernov theorem:

$h_{\mu}(T)=\frac{1}{n}h_{\mu}(T^n)$ .

Now we state the result of Margulis and Ruelle:

Let $M$ be a compact riemannian manifold, $f:M\to M$ is a diffeomorphism and $\mu$ is a $f$ -invariant measure.

Entropy is always bounded above by the sum of positive exponents;i.e.,

$h_{m}(f)\leq \int_{i}\lambda_i^{+}(x)dimE_i(x)dm(x).$

Where $dimE_i(x)$ is the multiplicity of $\lambda_i(x)$ and $a^{+}=max(a,0)$ .

Pesin show the inequality is in fact an equality if $f\in C^2$ and $m$ is equivalent to the Riemannian measure on $M$ . So this is also sometime known as Pesin’s formula.

F.Ledrappier and L.S.Young generate the result of Pesin.

One of their main result is:

$f:M\to M$ is a $C^2$ diffemoephism, where $M$ is a compact riemanian manifold, f is compatible with the Lesbegue measure on $M$ , and

$h_m({f,\mu})=\int_{M}\lambda_idim(V_i)dm$

If and only if on the canonical defined quation manifold $M/W_{\mu}$, i.e. the manifold mod unstable manifold $W_{\mu}$, the induced conditional measure $m_{\xi}$ is absolute continuous.

Remark: according to my understanding, the equality just mean in some sense we have the inverse estimate:

$h_{m}(f,\mu)\geq \int_{M}\lambda_idim(V_i)dm.$

This result maybe just mean near the fix point of $f$ ,i.e. the place charge the topology of the foliation, we have the inverse estimate. Such a inverse estimate will lead a control of the singularity of the push forward measure $m_{\xi}$ on the quation manifold. So $m_{\xi}$ have good regularity. But this idea is not complete to solve the problem.

Now we begin to get a geometric explain and which will lead a rigorous proof of the inequality:

$h_{m}(f)\leq \int_{i}\lambda_i^{+}(x)dimE_i(x)dm(x).$

At first we could observe that the long time average $\lim_{n\to \infty}\frac{1}{n}log||Df^n||$ of $Df$ could be diagonal. Assume after diagonal the eigenvalue is

$\lambda_1\leq \lambda_2\leq \lambda_3\leq...\leq \lambda_{n-1}\leq \lambda_n$ .

This eigenvalue could divide into 3 parts: <0,=0,>0.

This will lead to a direct sum decomposition of the tangent bundle $TM$ :

$TM\simeq E_{u}\otimes E_s\otimes E_c.$

Where $E_u$ is the part corresponding to the eigenvalue>0, For this part we consider the more refinement decomposition:

$E_u=\otimes_{k=1}^rV_k$ , $V_k$ is the eigenvector space of $\lambda_k$ . The dimension of $V_k$ is $dim V_k$.

On the other hand, we have a equality of metric entropy:

$h_{m}(f)=\frac{1}{n}h_{m}(f^n)=\sup_{\alpha\in partition \ set}\frac{1}{n}h_m(f^n,\alpha)$ .

For the later one, $\alpha$ is a measurable partition of $M$ , then $\alpha$ could always be refine to a smaller partition $\beta$ , and we have:

$h_{m}(f,\alpha)\leq h_{m}(f,\beta)$ .

Now we arrive the central place of the proof:

every partition could be refine by a partition with boundary of almost all cubes is parallel to the foliation. So we focus ourselves on the portion $\beta$ and all boundary of cubes in $\beta$ is parallel to the eigenvector.

Under this situation, we need only estimate the numbers of $\vee_{i=1}^nT^i\beta$ . Estimate it is not very difficult. we need only observe the following two thing:

1.

$\lim{n\to \infty}$ exists a.e. in $M$ . So this lead to the definition of foliation almost everywhere, and except a measurable zero set. In fact this set is the set of fix point of $M$ under $f$ .

2.

After a rescaling, every point which is not a fix point of $f$ could be understand as it is far away from fix points. Then the foliation could be understand as a product space locally. The flow with the direction which the eigenvalue is less than 1 cold not change $\vee_{i=1}^nT^i\beta$ . The direction with eigenvalue equal to 1 is just transition and just change the number of $\vee_{i=1}^nT^i\beta$ with polynomial growth. But the central thing is the direction with eigenvalue large than one and will make $\vee_{i=1}^nT^i\beta$ change with viscosity $e^{\lambda_i}$ . and we product it and get :

$h_{m}(f,\mu)\leq \int_{M}\lambda_i dim(V_i)dm$ .

In fact the proof only need $f$ to be $C^1$

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

metric entropy 衡量一个保测动力系统在可测意义下产生信息的速率。对光滑动力系统，它和 Lyapunov exponents 之间有深刻联系：正 Lyapunov 指数给出不稳定方向上的体积增长，而 entropy 记录可区分轨道的增长。

Ruelle 不等式和 Pesin 公式把 entropy 与正 Lyapunov 指数联系起来。

1. Entropy 的基本图像

给定有限 measurable partition $\mathcal P$，Shannon entropy 是

$$H_\mu(\mathcal P)=-\sum_{A\in\mathcal P}\mu(A)\log\mu(A).$$

迭代下的平均信息增长定义为

$$h_\mu(f,\mathcal P)=\lim_{n\to\infty}\frac1n H_\mu\left(\bigvee_{j=0}^{n-1}f^{-j}\mathcal P\right).$$

再对所有 partition 取上确界，得到 $h_\mu(f)$。

2. Spanning 与 separating

拓扑 entropy 可以用 $(n,\varepsilon)$-spanning set 或 separating set 描述。两个点若在前 $n$ 次迭代中始终很近，就被看作同一条轨道影子。entropy 记录为了覆盖所有轨道影子，需要多少个名字。

这个图像和 metric entropy 的 partition 定义相互对应：一个是拓扑尺度，一个是测度尺度。

3. Ruelle inequality

设 $f$ 是紧 Riemannian manifold 上的 $C^1$ diffeomorphism，$\mu$ 是 $f$-invariant measure。Ruelle 不等式说

$$h_\mu(f)\le \int \sum_{\lambda_i(x)>0}\lambda_i(x)m_i(x)\,d\mu(x).$$

右端是正 Lyapunov exponents 的总和。直观上，系统能产生的信息不可能超过不稳定方向的体积膨胀能力。

4. Pesin formula

在更光滑并且测度与 Riemannian volume 绝对连续的情形，Pesin 公式给出等号：

$$h_\mu(f)=\int \sum_{\lambda_i(x)>0}\lambda_i(x)m_i(x)\,d\mu(x).$$

这说明所有不稳定方向上的 expansion 都真正转化成了信息增长，没有被 singular conditional measures 损失掉。

5. Ledrappier-Young 的视角

Ledrappier-Young 理论进一步解释等号何时成立：关键在于 unstable foliation 上的 conditional measures 是否绝对连续。若条件测度太奇异，几何膨胀不一定变成可测 entropy；若条件测度足够连续，膨胀就能被 entropy 读出来。

因此 entropy、Lyapunov exponents 和 foliation 上的条件测度，实际上是同一件事的三个侧面。

2017年11月21日
Gromov 式尺度思想在抛物方程中的应用

旧博客原文

原题：Gromov’s idea applicate to parabolic equation

Gromov’s idea applicate to parabolic equation.

Key point:

1.rescaling+renormalization.

2.analysis it on every scale.

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

抛物方程的一个基本特征是空间和时间的尺度不同：若空间尺度缩小为 $r$，时间尺度就应缩小为 $r^2$。Gromov 式思想强调：不要只在一个固定尺度看方程，而要不断 rescale、renormalize，并在每个尺度上寻找紧性和极限模型。

抛物方程的自然尺度是空间 $r$、时间 $r^2$；正则性分析通常要在所有尺度上比较。

1. 抛物缩放

以热方程为例，若 $u_t-\Delta u=0$，定义

$$u_r(x,t)=u(x_0+rx,t_0+r^2t).$$

则 $u_r$ 仍满足同类方程。这个不变性说明，局部正则性问题可以被搬到单位尺度上处理。

2. 重整化

如果在某点附近梯度或曲率变大，就按最大量归一化。得到的序列若有紧性，极限通常是一个定义在全空间或半空间上的古老解。然后利用 Liouville theorem 排除非平凡极限。

3. 每个尺度的分析

尺度方法的关键不是一次缩放，而是在 dyadic scales 上反复比较：

$$Q_r(z_0)=B_r(x_0)\times(t_0-r^2,t_0).$$

若某个量在小尺度上不能衰减，就会产生 blow-up 极限；若 blow-up 极限被排除，就得到衰减估计。

4. 与几何分析的联系

Ricci flow、mean curvature flow 和非线性扩散方程中都可以看到这个结构。Gromov 风格的贡献在于把局部估计转成“所有尺度上的紧性与极限分类”。

2017年11月20日
Hilbert 第十六问题笔记：limit cycle、奇点与拓扑图像
旧博客原文

原题：Hilbert 16th problem

Introduction

the statement of Hilbert’s 16th problem:

$H(n)<\infty?$

definition of $H(n)=max$

Limit cycle:

Try beginning with Bendixon-Poincaré theorem, which is classical stuff and belongs to a lot of textbooks on vector fields.

Affine invariance

The number of limit cycle is invariant under affine map.

Classification of singular point

Bezout theorem

Example

1. $\frac{dx}{dt}=y,\frac{dy}{dt}=x$ .The graph is just like:

2. $\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=x-y$ .The graph is just like:

3. $\frac{dx}{dt}=x^2-y^2,\frac{dy}{dt}=5-y$ .The graph is just like:

4. $\frac{dx}{dt}=x^2-y^2,\frac{dy}{dt}=-y$ .The graph is just like:

5. $\frac{dx}{dt}=x^3-y^3,\frac{dy}{dt}=5-y$ .The graph is just like:

6. $\frac{dx}{dt}=y^2-x^2+1,\frac{dy}{dt}=y$ . The graph of it is just like:

Now we try to explain the phenomenon we see. At first we can see there is no limit cycle in the picture. The bifurcation place is just the place $\frac{dx}{dt}=0\ or \frac{dy}{dt}=0$ and is just like 3 lines. And there are two singularity $(-1,0),(1,0)$ .

7. $\frac{dx}{dt}=y,\frac{dy}{dt}=y-x^2-x^2y$

8. $\frac{dx}{dt}=y,\frac{dy}{dt}=y-x-x^2y$ :

9. $\frac{dx}{dt}=y(y-3),\frac{dy}{dt}=(y-3-x-x^2y)(y-x-x^2y)$ :

exist 2 limits cycles.

10. $\frac{dx}{dt}=y(y-3)(y-5),\frac{dy}{dt}(y-5-x-x^2y)(y-3-x-x^2y)(y-x-x^2y)$

Tree structure

In general, the lower bound of $H(n)$ is established. first $H(n)=O(n^2)$ by Otrokov,and later proved to be $H(x)=O(ln(n)n^2)$ .

If right, this upper bound estimate is combine of two things:

1.every limit cycle do not intersect.

2.every unclear limit cycle contain a singularity.

Rough strategy attack Hilbert 16th problem

Step1:

The first step is to do some simplify, we know the number of limit cycles do not change under a affine map $(x,y)\to (\hat x,\hat y)=(x,y)A$ , where A is a nonsingular 2*2 matrix. So we could classify the topological graph of the dynamic system.

Step2:

Classification the singularity under affine map, and investigate the topological graph of the topological graph of the singularity by floer cohomology. There is only finite type and we could focus on them one by one.

Every color in the graph is an area $P,Q$ do not have same component in it. And the boundary of areas is just the same component of $P,Q$ if it exists.

Step3:

Bezout theorem tell us if two polynomials $P(x,y), Q(x,y)$ do not have same connected component then the intersection $I_{P,Q}\leq deg(P)deg(Q)$ . And we can divide the space $R^2$ into finite parts, $P(x,y),Q(x,y)$ restrict to every part do not have the same component. And we have a upper bound control on the number of part when $\max\{deg(P(x,y)),deg(Q(x,y))\} \leq n$ . A easy arrive bound could be $4^n$ .

Step4:

Now we focus ourself on 1 part where $P(x,y),Q(x,y)$ do not have same component on it. Now we begin to proof there will be a relationship between the limit cycle, very like a tree structure, it will combine with the following two thing:
1. Every limit cycle contains at least one singularity or one smaller limit cycle inside it.
2. Every pair of limit cycle $(A,B)$ , $A,B$ are limit cycle and $A$ is inside of $B$ , then there is at least one singularity or another limit cycle contain in $\Omega$ .

This kind of topological result will lead to a upper bound of number of limit cycle and end the proof of Hilbert 16th problem.

Planar polynomial vector field for a harmonic pair of polynomials

In this case you can consider the heat equation $\partial_t u(z,t)=\Delta u(z,t)$ . If the number of the limit cycle change, it must be the time to pass a singularity. and take $t=\infty$ , the dynamic system coverage to a very simlpe one and in particular it do not have limit cycle. So we need only look at the moments passing singularity.

Has the system of ODEs:

$\frac{dx}{dt}=P(x,y)\\ \frac{dy}{dt}=Q(x,y)$

been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of a holomorphic polynomial $F=F(z)$ , $z=x+iy$ ?

I am looking to learn a bit about (complex) ODEs and their interplay with algebraic geometry by some examples, but I couldn’t find anything on this special case in Ilyashenko’s survey on Hilbert 16 (I guess this case is too special and/or not very interesting as far as Hilbert 16 is concerned).

Nontheless, it seems very natural. If we set $\gamma(t)=x(t)+iy(t)$ , this amounts to the equation $latex \int_{\gamma_t}\frac{dz}{F(z)}=t$
where $\gamma_t$ is the curve $\gamma$ “truncated” at $t$ and the RHS is in particular **real**. This can be taken further, for example by assuming $\gamma$ is closed and using the residue theorem to obtain constraints on (the coefficients of) $F$ .

First, this case is totally uninteresting regarding Hilbert XVI. Indeed, there are no limit cycles in such systems. The $\alpha / \omega$ -limit of a trajectory is either a point or a non-isolated cycle (center case).

A singularity at $a\in \mathbb C$ (*i.e.* a root of $F$) can only be of three types, according to the value of $F'(a)$ :

1. Source/focus: $F'(a)\notin i\mathbb R$ .
2. Center: $F'(a)\in i\mathbb R_{\neq 0}$ .
3. Flower with $2k$ petals: $F'(a)=0$ with multiplicity $k$ .

In addition there is a pole at infinity (if $\deg(F)>0$ ) with exactly $2\deg(F)$ separatrices, reaching the singularity in finite time. The bassins of attraction / center regions attached to the above singularities are delimited by the separatrices.

[![enter image description here][2]][2]

S. Smale began to get interested in the question in the early 80’s while laying the foundations for BSS computational model (*The fundamental theorem of algebra and complexity theory*, 1981). He proposed a numerical root solver for polynomials by following the flow of $\frac{F}{F'}$ . This started some works on the topic, for instance by Schub, Tischler, William (*The Newtonian graph of a complex polynomial*, 1988) or Benzinger (*Plane autonomous systems with rational vector fields*, 1991)…

In the case of these vector fields, the topological class is entirely encoded by their Newtonian graph (or the «dual» spinal graph) given by the incidence graph of the $\alpha / \omega$ -limits of trajectories (in red on the picture). The main result for polynomials is that it is a tree. See *e.g.* Sverdlove (*Inverse problems for dynamical systems*,1981) and Schecter, Singer (*A class of vectorfields on $\mathbb S^2$ that are topologically equivalent to polynomial vectorfields*,1985) and Jongen, Jonker, Twilt (*On the classification of plane graphs representing structurally stable rational Newton flows*,1991).

The conformal classification has been initiated by Douady, Estrada and Sentenac (unpublished monograph, 2005) for the generic case (only focus/source singularities) and completed by Branner and Dias (*Classification of complex polynomial vector fields in one complex variable*, 2010). In addition to the combinatorial (topological) invariant, a complex «time-shift» (related to the integrals $\int_\gamma\frac{1}{F(z)} dz$ ) is associated to the separatrices, providing a complete conformal invariant.

In that latter context, the function $\int\frac{1}{F(z)} dz$ is called a Fatou coordinates. It is a rectifying chart for the vector field, and has many interesting dynamical properties.

Notice also the deep and beautiful relationship between spinal graph and *Dessins d’enfants*, as established by Pilgrim (*Polynomial vector fields, dessins d’enfants, and circle packings*,2006), related to [this question](https://mathoverflow.net/questions/118527).

Reference

Classification of Singularities and Bifurcations of Critical Points of Even Functions

E.A.Kudryavtseva, E.Lakshtanov

Classification of the singularity in even degree case.

Adjoint harmonic case have been studied. Look into the recent paper
Langley, J. K. Trajectories escaping to infinity in finite time. Proc. Amer. Math. Soc. 145 (2017), no. 5, 2107–2117, and the reference list in this paper.

They were also studied by physicists:

Bender, Carl M.; Hook, Daniel W.
Complex classical motion in potentials with poles and turning points.
Stud. Appl. Math. 133 (2014), no. 3, 318–336.

EDIT. I forgot to mention this:

B. Branner, K. Dias, Classification of complex polynomial vector fields in
one complex variable, Journal
Journal of Difference Equations and Applications
Volume 16, 2010 – Issue 5-6:

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Hilbert 第十六问题的第二部分问：平面多项式向量场的 limit cycles 数量能否只用次数控制？这看起来像一个拓扑问题，因为 limit cycles 是平面轨道的闭曲线；但真正困难在于解析和代数几何结构如何控制这些闭轨道的产生与消失。

Hilbert 第十六问题把 limit cycles、奇点、separatrix graph 和 return map 零点数放到同一个问题里。

1. 基本对象

考虑平面多项式系统

$$\dot x=P(x,y),\qquad \dot y=Q(x,y),$$

其中 $P,Q$ 是次数不超过 $d$ 的多项式。limit cycle 是孤立的周期轨道。Hilbert 第十六问题想问是否存在一个只依赖 $d$ 的上界 $H(d)$，控制所有这类系统的 limit cycles 数量。

对 $d=1$，线性系统没有孤立 limit cycles。对 $d\ge2$，问题迅速变得非常困难；一般情形至今仍未完全解决。

2. Poincare-Bendixson 的图像

Poincare-Bendixson theorem 告诉我们，平面系统的紧 $\omega$-limit set 若没有奇点，通常会落到周期轨道上。这使得二维动力系统比高维系统更有拓扑可视化：轨道、奇点、separatrices 和 limit cycles 共同组成一张平面图。

但可视化不等于容易。limit cycle 可以嵌套，可以从 polycycle bifurcation 中产生，也可能在参数变化时通过奇点附近的精细结构出现。

3. Affine invariance 与奇点分类

非退化 affine 变换不会改变 limit cycles 的数量。因此可以尝试先把系统化到较简单的坐标，再分类奇点局部模型。奇点由

$$P(x,y)=Q(x,y)=0$$

给出。若 $P,Q$ 没有公共因子，Bezout theorem 给出复射影意义下交点数不超过 $d^2$。这给了奇点数量的粗上界。

但是 limit cycles 不只由奇点数量决定。一个奇点周围可能有复杂的 separatrix structure；多个奇点之间的连接也可能产生 bifurcation。

4. 树结构的想法与局限

一个自然设想是把嵌套的 limit cycles 看成树：外层 cycle 包含内层 cycle 或奇点；两层之间的 annulus 里如果没有奇点，也许能排除新的孤立周期轨道。这样的拓扑图像确实有启发性。

困难在于，排除一个 annulus 中的周期轨道需要 Dulac function、Abelian integral、return map 或更精细的解析控制。拓扑结构给出框架，真正的上界需要估计 Poincare return map 的零点数。

5. Harmonic pair 的特殊情形

若 $P$ 和 $Q$ 是某个 holomorphic polynomial $F(z)$ 的实部和虚部，即

$$P+iQ=F(z),$$

则系统具有复分析结构。此时很多 Hilbert 第十六问题中的困难现象会消失：轨道可以通过

$$\int \frac{dz}{F(z)}=t$$

来理解，奇点由 $F$ 的零点控制。这样的系统通常不会给出 Hilbert XVI 中真正困难的孤立 limit cycles；它更像是一个用来学习复 ODE、Newtonian graph 和代数拓扑图像的模型。

6. 这条路线的意义

一个可能的攻击路线是：先用 affine 变换和 Bezout 控制奇点类型；再用 separatrix graph 把平面分解成有限区域；最后在每个区域中估计 return map 或 Dulac integral 的零点。这个策略很自然，但每一步都需要强解析输入。

Hilbert 第十六问题之所以难，正是因为它站在三件事的交界处：平面拓扑告诉我们轨道如何嵌套，代数几何控制多项式奇点，分析估计决定 limit cycles 能否真正出现。
2017年11月18日
一个 determinantal formula：Gram identity、Andreief identity 与对称多项式

旧博客原文

原题：A determinantal formula

I see a similar formula I wish to be true and merely have a proof in mind occur as a MO’s problem:

In my research, I encounter the following formula which I believe is correct (checked for $n\le3$ ). Is it classical ?

I am given a real symmetric matrix
$S:=\int Y(t)Y(t)^Td\mu(t),$
where $\mu$ is a probability and $Y(t):\Omega\rightarrow{\mathbb R}^n$ .

Let $\sigma_k(S)$ be the elementary symmetric polynomial in the eigenvalues of $S$ . For instance, $\sigma_1(S)$ is the trace and $\sigma_n(S)$ the determinant. The following formula gives $\sigma_k(S)$ in terms of the Gram matrix $G_k(s_1,\ldots,s_k)$ whose entries are the scalar products $Y(s_i)\cdot Y(s_j)$ .

$\sigma_k(S)=\frac1{k!}\int^{\otimes k}\det G_k(s_1,\ldots,s_k)\,d\mu(s_1)\cdots d\mu(s_k).$

Remark that $S$ is positive semi-definite. The integrand is non-negative, as well as $\sigma_k(S)$ . The integrand vanishes identically iff $Y(t)$ takes values in a subspace of dimension $<k$ , which is the condition under which $\sigma_k(S)$ vanishes. It follows that, if the formula above failed, it would be because of an inequality between strictly positive numbers.

The case $k=n$ is a consequence of the identity

$\int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)$

which I have seen under the names “Andreief identity” and also “Gram identity”. The proof is elementary using the Leibniz formula for the determinant.

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

很多行列式公式可以理解为 Gram identity 或 Andreief identity 的变体。它们把矩阵的对称多项式、向量组的体积和积分中的 determinant 联系起来。

Gram identity 和 Andreief identity 把行列式、体积平方和对称多项式联系起来。

1. Gram matrix

给定向量 $v_1,\ldots,v_m$，Gram matrix 为

$$G_{ij}=\langle v_i,v_j\rangle.$$

$\det G$ 等于这些向量张成平行体体积的平方。因此 $G$ 半正定，且 $\det G=0$ 当且仅当向量线性相关。

2. Andreief identity

Andreief identity 的典型形式是

$$\int \det(f_i(x_j))\det(g_i(x_j))\prod_j d\mu(x_j)
=n!\det\left(\int f_i(x)g_j(x)d\mu(x)\right).$$

它是积分版的 Cauchy-Binet 公式。

3. 对称多项式

若 $A$ 是半正定矩阵，特征值为 $\lambda_1,\ldots,\lambda_n$，第 $k$ 个 elementary symmetric polynomial 是

$$e_k(A)=\sum_{i_1<\cdots
它也等于所有 $k\times k$ 主子式之和。

4. 几何解释

$e_k(A)$ 可以看成 $A$ 在所有 $k$ 维方向上的体积膨胀总和。若 $A$ 来自随机向量的 Gram matrix，那么对应公式会把 $e_k(A)$ 写成某种 determinant 的积分平均。

5. 证明方式

这类公式通常从 Leibniz determinant expansion 出发，交换求和和积分，再识别为 Cauchy-Binet 或 Andreief identity。核心不是计算技巧，而是“行列式等于体积平方”的几何含义。

2017年11月17日
等周不等式：Steiner 对称化、Minkowski-Steiner 公式与变分图像

旧博客原文

原题：Isoperimetric inequality

Introduction

the statement go isometry inequality is very simple:

$\Omega\subset R^n$ , iff $\Omega$ is a ball, $\frac{Vol(\Omega)}{Surf(\Omega)}$ arrive a minimum .

This is a classical problem in variation theory. The difficult is divide into two parts. The first is to create a “flow” which descrement the energy and the “flow” is compatible with the feature of a ball, i.e. every set under the flow will tend to like a “ball”. The second one is to proof there exist a unit in the space $surf(\Omega)=constant>0$ make the Energy $E(\Omega)=Vol(\Omega)$ arrive a minimum.

Combine this two property we can consult that ball is the set and definitely the only set make the $\frac{Vol(\Omega)}{Surf(\Omega)}$ arrive the minimum.

first difficulties

The energy $E(\Omega)$ is scaling invariance. The first difficult could divide into two part:

Restrict to convex set

the first is to deform a set into a convex set and proof this process would not lower $\frac{Vol(\Omega)}{Surf(\Omega)}$ . This could been a little subtle. and the way I image could make sense is just like the following transform:

but this process is harder in higher dimension, for example:

Convex set to a ball

Steiner symmetric process.

Affine transform

Minkowski–Steiner formula
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the “derivative” of enclosed volume in an appropriate sense.

The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.

Statement of the Minkowski-Steiner formula

Let $n \geq 2$ , and let $A \subsetneq \mathbb{R}^{n}$ be a compact set. Let $\mu (A)$ denote the [[Lebesgue measure]] (volume) of $A$ . Define the quantity $\lambda (\partial A)$ by the ”’Minkowski–Steiner formula”’:

$\lambda (\partial A) := \liminf_{\delta \to 0} \frac{\mu \left( A + \overline{B_{\delta}} \right) - \mu (A)}{\delta}$

where:

$\overline{B_{\delta}} := \left\{ x = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} \left| | x | := \sqrt{x_{1}^{2} + \dots + x_{n}^{2}} \leq \delta \right. \right\}$

denotes the [[closed ball]] of [[radius]] $\delta > 0$ , and:

$A + \overline{B_{\delta}} := \left\{ a + b \in \mathbb{R}^{n} \left| a \in A, b \in \overline{B_{\delta}} \right. \right\}$

is the [[Minkowski sum]] of $latexA$ and $\overline{B_{\delta}}$ , so that:

$A + \overline{B_{\delta}} = \left\{ x \in \mathbb{R}^{n} | |x - a| \leq \delta \mbox{ for some } a \in A \right\}$ .

Surface measure

For “sufficiently regular” sets $A$ , the quantity $\lambda (\partial A)$ does indeed correspond with the $(n - 1)$ -dimensional measure of the [[boundary (topology)|boundary]] $\partial A$ of $A$ . See Federer (1969) for a full treatment of this problem.

Convex sets

When the set $A$ is a [[convex set]], the [[limit inferior|lim-inf]] above is a true [[Limit of a sequence|limit]], and one can show that

: $\mu \left( A + \overline{B_{\delta}} \right) = \mu (A) + \lambda (\partial A) \delta + \sum_{i = 2}^{n - 1} \lambda_{i} (A) \delta^{i} + \omega_{n} \delta^{n}$ ,

where the $\lambda_{i}$ are some [[continuous function]]s of $A<$ (see [[quermassintegral]]s) and $\omega_{n}$ denotes the measure (volume) of the [[unit ball]] in $\mathbb{R}^{n}$ :

: $\omega_{n} = \frac{2 \pi^{n / 2}}{n \Gamma (n / 2)}$ ,

where $\Gamma$ denotes the [[Gamma function]].

==Example: volume and surface area of a ball==

Taking $A = \overline{B_{R}}$ gives the following well-known formula for the surface area of the [[sphere]] of radius $R$ , $S_{R} := \partial B_{R}$ :

: $\lambda (S_{R}) = \lim_{\delta \to 0} \frac{\mu \left( \overline{B_{R}} + \overline{B_{\delta}} \right) - \mu \left( \overline{B_{R}} \right)}{\delta}$
:: $= \lim_{\delta \to 0} \frac{[ (R + \delta)^{n} - R^{n} ] \omega_{n}}{\delta}$
:: $= n R^{n - 1} \omega_{n}$ ,

where $\omega_{n}$ is as above.

something with more details

The second difficulties

To establish a continue property of the Energy functional $E(\Omega)= \frac{Vol(\Omega)}{Surf(\Omega)}$ .

the continuous property is consider with all open set $\Omega$ with Gromov-Hausdorff metric $d(\Omega_1,\Omega_2)= \inf_{metric\ d on \Omega_1 \cup \Omega_2}\sup_{x_1\in \Omega_1, x_2\in \Omega_2}d(x_1,x_2)$ .

We need to proof the continuous of $E(\Omega)$ with the Gromov-hausdorff metric on the space consist with convex open sets.

To remark,we need to observe that polygon approximation is just corresponding to the $\delta-seperate$ points approximation in Gromov-hausdorff distance. and definitely carefully refinement of this kind of approximation could lead to the result of continuous of the energy $E(\Omega)$ on convex set.

A second remark, we definitely need a definition of the $surf(\Omega)$ it could be achieve with open convex set $\Omega$ by a outer and inter approximation by polygon and the error term estimate.

Further remark, isoperimetric inequality is a general phenomenon.

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

等周不等式说，在给定体积的集合中，球的边界面积最小。它是变分法、凸几何和几何测度论共同的基本定理。

等周不等式说明固定体积下球最小化 perimeter，可由对称化或 Brunn-Minkowski 思想证明。

1. 基本陈述

在 $\mathbb R^n$ 中，等周不等式为

$$P(\Omega)^n\ge n^n\omega_n |\Omega|^{n-1},$$

等号当且仅当 $\Omega$ 是球。这里 $P(\Omega)$ 是 perimeter，$\omega_n$ 是单位球体积。

2. 变分图像

一个自然想法是构造能量下降 flow，使任意集合逐渐变圆，并且不增加 perimeter、不改变体积。真正困难在于：一般集合可能很粗糙，flow 的存在性和紧性都需要处理。

3. Steiner 对称化

Steiner symmetrization 沿一个方向把每条平行线上的截面替换成居中的区间。它保持体积，并且不增加 perimeter。重复对称化后，集合越来越接近球。

这提供了一个几何证明路线：先把集合变得越来越对称，再用紧性取极限。

4. Minkowski-Steiner 公式

对足够好的集合，考虑外平行体

$$\Omega_t=\Omega+tB.$$

Minkowski-Steiner 公式把 $|\Omega_t|$ 展开成 $t$ 的多项式，其一次项与 perimeter 相关：

$$\left.\frac{d}{dt}\right|_{t=0}|\Omega+tB|=P(\Omega).$$

配合 Brunn-Minkowski 不等式，可以推出等周不等式。

5. 核心困难

等周问题的困难可以分成两部分：一是证明极小值存在，二是证明极小者只能是球。前者需要紧性和 lower semicontinuity，后者需要对称化、Euler-Lagrange 方程或 Brunn-Minkowski 凹性。

2017年11月16日
k-Hessian 方程与 k-curvature 方程：椭圆性、测度与 Wolff 势

旧博客原文

原题：k-hessian equation and k-curvature equation

here is the problem, how to understand k-hessian equation and k-curvature equation.

k-hessian equation

k-hessian equation is:

$H_k(u)=\sigma_k(D^2(u))=f$ (*)

where u is admissible, i.e. $\forall 1\leq i\leq k$ , $\sigma_i(D^2(u))\geq 0$ . this is just the condition to make (*) be a elliptic equation.

The most important result is the following three:

1.sovable (*) with direchlet boundary condition.

This is mainly the contribution of Caffaralli in 90’s. According flexible function and maximum principle we can establish the $C^{1,\alpha}$ estimate and $C^{2,\alpha}$ estimate in the inter. And the $C^{2,\alpha}$ estimate near the boundary is establish according to the conformation invariant and some perbutation of the solution of k-hessian equation after special rescaling.

2.Hessian measure.

This is mainly the work of X.J.Wang and Trudinger. they proved:

in the meaning of viscosity solution, if $\sigma_k(D^2(u))=f$ . then we can associate a measure $\mu$ with $u$ ,and the following is right:

when $u\in C^2(\Omega)$ , $\mu(B_r(x))=\int_{B_r{x}}\sigma_k(D^2(u))$ .

if $u_1,..,u_n,...$ coverage to $u$ . then $\mu_1,...,\mu_n,...$ coverage to $\mu$ in weak sense.

this is merely depend on a priori estimate on $u$

3.pointwise estimate corresponding wolff potential.

the Wolff potential is:

$W^{\mu}_{k}(x,r)= \int_{0}^r(\frac{\mu(B_t(x))}{t^{n-2k}})^{\frac{1}{k}}\frac{1}{t}dt$

We can easily use rescaling to understand the reasonable of this potential, and use this potential Lubutin establish the following pointwise estimate:

$u\in \Phi_k(B_{4R}(x))$ , $u\leq 0$ , then we have:

$W^{\mu}_k(x,\frac{R}{2})\leq |u(0)| \leq W^{\mu}_k(x,2R)-sup_{B_{2R}}|u|$

the RHS could look as a corollary of classical A-B-P estimate. the LHS need combine several observation. mean-value property and some else.

This result could use to establish some result on singularity point can be removable.

k-curvature equation

1.sovable (*) with direchlet boundary condition.

This is also established by cafferalli.

2.curvature measure.

This is established very recently. mean curvature equation in 2014, by perron lift and modified, general case in 2016 by more complex calculate and method.

3.pointwise estimate corresponding wolff potential.

This still do not established, and is the main thing I focus on. Due to we can look as k-curvature as a “projection” of k-hessian equation, Calderon-Zegmund decomposition and the estimate of k-hessian equation maybe useful.

My ideas

look is as “average” of “loop space”, “surface space”.

1.Grassmannian bundle

n algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
$p:G_{d}(E)\to X$
such that the fiber

$p^{-1}(x)=G_{d}(E_{x})$ is the Grassmannian of the d-dimensional vector subspaces of $E_x$ . For example,

$G_{1}(E)=\mathbb {P} (E)$ is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into

$0\to S\to p^{*}E\to Q\to 0$ .
Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank $r = rk(E)$ and

$\wedge ^{r}S$ is the determinant line bundle. Now, by the universal property of a projective bundle, the injection

$\wedge ^{r}S\to p^{*}(\wedge ^{r}E)$ corresponds to the morphism over X:
$G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)$ ,
which is nothing but a family of Plücker embeddings.

The relative tangent bundle $T Gd(E)/X$ of $Gd(E)$ is given by[1]
$T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,$
which is morally given by the second fundamental form. In particular, when d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:
$0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0$ ,
which is the relative version of the Euler sequence.

2.Explain of the fully nonlinear elliptic equation

Now, we could consider the determination $\sum_{i_1,...,i_k\in\{1,...,n\}}det(u_{ij})_{i,j\in \{i_1,...,i_k\}\times\{i_1,...,i_k\}}$ as the determination of transform: $(u_{i_1},...,u_{i_k}) \longrightarrow (e_{i_1},...,e_{i_k})$ .

Now we need to understand $\sigma_k(D^2(u))=f$ at a point $x_0$ as the average of determination of transform matrix of $(u_{i_1},...,u_{i_k}) \longrightarrow (e_{i_1},...,e_{i_k})$ on Grassmannian manifold $G_k(x_0)$ is equal to $f(x_0)$ , i.e.:

$\int_{G_k(x_0)} det(\frac{\partial u_{i_a}}{\partial e_{i_b}}) d\mu=f(x_0)$

where $\mu$ is the natural haar measure on $G_k(x_0) \simeq G_k$ .

But the difficult to make the argument rigorous is that $u_i$ is scale and $e_i$ is vector.

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

$k$-Hessian 方程是完全非线性椭圆方程中的基本模型。它把 Hessian 矩阵特征值的第 $k$ 个基本对称函数作为主算子；椭圆性不再自动成立，而是依赖 admissible cone。

$k$-Hessian 方程的椭圆性来自 admissible cone，弱解理论则进入 Hessian measure 和势估计。

1. 方程与 admissibility

设 $\lambda(D^2u)$ 是 Hessian 的特征值，定义

$$S_k(D^2u)=\sigma_k(\lambda(D^2u)).$$

$k$-Hessian 方程写作

$$S_k(D^2u)=f.$$

为了让方程椭圆，需要要求

$$\lambda(D^2u)\in \Gamma_k=\{\sigma_1>0,\ldots,\sigma_k>0\}.$$

这样的 $u$ 称为 $k$-admissible。这个条件是非线性椭圆理论的入口。

2. Dirichlet 问题

Dirichlet 问题要求

$$S_k(D^2u)=f\quad\text{in }\Omega,\qquad u|_{\partial\Omega}=\varphi.$$

经典策略是建立 $C^0$、梯度和二阶先验估计，再用连续性方法。边界估计通常最细，需要利用 domain 的几何条件、barrier function 和 rescaling。

3. Hessian measure

对非光滑 admissible 函数，也可以定义 Hessian measure。若 $u_j\to u$，并且 $u_j$ 是光滑 admissible，那么在合适条件下

$$S_k(D^2u_j)\,dx \rightharpoonup \mu_k[u].$$

这把方程扩展到 viscosity/pluripotential 风格的弱解框架。它类似 Monge-Ampere measure，但 $k$-Hessian 的 cone 结构更复杂。

4. Wolff potential

点态估计中会出现 Wolff potential：

$$W_{\alpha,p}^\mu(x)=\int_0^\infty\left(\frac{\mu(B(x,r))}{r^{n-\alpha p}}\right)^{1/(p-1)}\frac{dr}{r}.$$

它描述右端测度在不同尺度上的集中。对 $k$-Hessian 方程，解的上下界可以用相应的 Wolff potential 控制。这个形式可以从 scaling 看出：非线性阶数决定了势函数的指数。

5. 与几何曲率方程的联系

$k$-curvature 方程通常把 hypersurface 的第 $k$ 个曲率函数固定下来。解析上它和 Hessian 方程共享 symmetric polynomial、admissible cone 和 fully nonlinear ellipticity。几何问题中的正则性，往往依赖同一套先验估计和弱测度理论。

2017年11月14日
Vinogradov mean value theorem：矩估计、Weyl sums 与 decoupling

旧博客原文

原题：Note on Vinogradov main theorem

1.Introduction

Question:
Vinogradov mean value
Let $k,s\in \mathbb N$ , $x\in R^k$ .

$J_{s,k}(N)=|\{(n_1,...,n_s,n_{s+1},...,n_{2s})|n_1^j+...+n_s^j=n_{s+1}^j+...+n_{2s}^j) \forall 1\leq j\leq k,1\leq n_i\leq N(1\leq i\leq s) \}|$
How to estimate $J_{s,k}(N)$ ?

We assume $f_{k}(x,N)=\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)$ , then by following clear calculate:

$\int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1...dx_k =\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_k &=\int_{[0,1]^k}\sum_{1\leq n_1,...,n_{2s}\leq N}e^{2\pi i[(n_1+...+n_{2s})x_1+...+(n_1^k+...+n_{2s}^k)x_k-(n_{s+1}+...+n_{2s})x_1-...-(n_{s+1}^k+...+n_{2s}^k)x_k]} &=|\{(n_1,...,n_{2s})| n_1^j+...+n_s^j=n_{s+1}^j+...+n_{2s}^j,\forall 1\leq j\leq k\}|$

we have:
$J_{s,k}(N)=\int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1...dx_k$
main conjecture:
$\forall \epsilon >0$ ,we have:
$J_{s,k}(N)<<N^{\epsilon}(N^s+N^{2s-\frac{1}{2}k(k+1)})$

theorem(Bourgain-Demeter-Guth)
Main conjecture hold in general.

2.Application

We have following directly application:

1.Waring problem

2.Bound Weyl sums.\

3.Zero-free region for Riemann-zeta function.

3.Relate to the decoupling theorem

Now we discuss the decoupling theorem. This theorem describe the phenomenon when we are considering the “expension” operator $E_{[0,1]}(g)$ cut off $E_{[0,1]}(g)$ into a lot of small boxes $E_{J}(g)$ , then the $L_{d(d+1)})$ norms of the operator could be bounded very well, in fact it is near orthonagonal.

[B-D-G]
Let $d\geq 2$ ,$0<\delta\leq 1$. Then for each ball $B\subset R^d$ of radious at least $\delta^{-d}$ .
$||E_{[0,1]}g||_{L^{d(d+1)}(w_B)}<< \delta^{-\epsilon}(\sum_{J\subset [0,1],|J|=\delta}||E_Jg||^2_{L^{d(d+1)(w_B)}})^{\frac{1}{2}}$
( $J$ runs over a partition of $[0,1]$ in $\delta$ -intervals)

Discretized version:
Now we discuss the discretization of decoupling type result. We could establish a relationship between the decoupling theorem and Vinogradov mean theorem. look at the sum:
$\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_n$
This could be view as a $2s$ norm of a constant function $h=1$ , with a lebergue measure $d\sigma$ on curve $\Gamma=\{(t,t^2,...,t^d):0\leq t\leq 1\}$ . this curve $\Gamma$ could be view as a canonical curve with non-vanish guess curvature.
$||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^{k}}|\int_{\Gamma}h(t,t^2...,t^n)e(tx_1+...+t^kx_k)|^{2s}d\sigma$

this is very similar with the restriction theorem:

[restriction theorem]
let $\Gamma$ be $n-1$ dimension parabolic in $R^n$ , then guess curvature of $\Gamma$ is non-vanish. $\sigma$ is a natural induced lebergue measure on $\Gamma$ , we have, for suitable exponents $p,p'$ come from rescaling arument.
$||\widehat{gd\sigma}||_{p'}\lesssim ||g||_p$

So it seems like these are the same thing, but unfortunately they are not,there are two things distinct them:
1.the density is defferent, it is a discrete sum in:
$\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_n$
but a continue integral in:
$||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^k}|\int_{\Gamma}h(t,t^2...,t^n)e(tx_1+...+t^kx_k)|^{2s}d\sigma$
so we need to construct a rescaling way to make the discretization one coverage to the continue one.a suitable fexiable function seems like $(w_B,B_{r}(c_B)),w_B(x)=(1-\frac{|x-c_B|}{R})^{-100k}$
2.there

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Vinogradov mean value theorem 控制 Weyl sums 的高阶矩，是 Waring problem、指数和估计和 zeta 函数零点区域中的基础工具。Bourgain-Demeter-Guth 用 decoupling theorem 证明了主猜想。

Vinogradov mean value theorem 把 Weyl sums 的高阶矩与 moment curve decoupling 联系起来。

1. Mean value

设

$$S(\alpha)=\sum_{n\le N}e(\alpha_1n+\alpha_2n^2+\cdots+\alpha_kn^k).$$

Vinogradov mean value 研究

$$J_{s,k}(N)=\int_{[0,1]^k}|S(\alpha)|^{2s}\,d\alpha.$$

它也等于某个 Diophantine system 解的个数。

2. 主猜想

主猜想断言

$$J_{s,k}(N)\lesssim_\varepsilon N^\varepsilon\left(N^s+N^{2s-k(k+1)/2}\right).$$

两个项分别对应 diagonal solutions 和维数计数给出的主项。

3. 应用

这个估计直接用于 Waring problem，也给出 Weyl sums 的强上界。通过指数和控制，可以进一步进入 zeta 函数零点区域和等分布问题。

4. Decoupling 视角

考虑 moment curve

$$\gamma(t)=(t,t^2,\ldots,t^k).$$

decoupling theorem 描述 extension operator 在小区间分解后的 $L^p$ 几乎正交性。离散化后，它与 Vinogradov mean value theorem 精确相连。

5. 思想总结

Vinogradov mean value 把数论中的方程计数、调和分析中的 Fourier extension、以及几何中的曲率结构放到同一个问题里。这是现代解析数论和 decoupling 理论交汇的代表。

2017年11月13日
从 linear 到 multilinear：Cauchy-Schwarz、分解与复杂度下降

旧博客原文

原题：Linear to Multi-linear

The technique that transform a problem which is in a linear setting to a multilinear setting is very powerful.

such like:

1.The renormalization technique in complex dynamic system, and the generalization

this is mainly the Ostrowoski representation,and something else.

2.Fouriour analysis

this can be view when it is difficult to investigate a quality about a function $f$ , it is always easier to take charge with some some part of $f$ , in this case is given by $\hat f(\xi),\xi \in R$ or $\hat f(k),k\in Z$ like cut $f$ into a lot of small parts,deal with every part and use some inequality(always the triangle inequality or similar thing) to glue it into a whole estimate of the quantity of $f$ .

3.Multi-scales theory

this is used in the improve of Minkowski dimension of 3-dim kakeya set by Katz-Tao.

4.The proof of Bourgain-Sarnak-Ziegler theorem

Theorem(B-S-Z). Let $F : N \to C$ with $|F| \leq 1$ and let $\nu$ be a multiplicative function with $|\nu| \leq 1$ . Let $\tau > 0$ be a small parameter and assume that for all primes $p_1, p_2 \leq e^{1/\tau}$ , $p_1 \neq p_2$ , we have that for $M$ large enough

$|\sum_{m\leq M} F(p_1m)\overline {F(p_2m)}| \leq \tau M$ .

Then for N large enough

$|\sum_{m\leq M} \nu(n)F(n) | \leq 2 \sqrt{\tau log(\frac{1}{\tau})}M$ .

this theorem is not difficult to prove by bilinear method and Cauchy-Schwarz,you can see the detail in https://arxiv.org/abs/1110.0992v1.
According to this theorem,to get a good approximation of $\sum_{1\leq k\leq x}\mu(k)f(k)$ we use need a good approximation on $\sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ .this will be much easier.but for the RHS $f(x)$ is very complicated so I do not have a non-trivial estimate for $\sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ until now.

5.The multiplier restriction theorem(Tao)

6.Some special construction in additive Combitriocs

Such like when we want to consider some set $A\subset Z_1$ satisfied $\frac{|A-A|}{|A+A|}>>1$ it is convenient to consider in a high dimensional linear space $Z^N$ rather than in $Z$ .

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

很多问题直接估计一个线性量很难，但把它平方、分解或转成多线性形式后，结构反而清楚。这就是从 linear 到 multilinear 的基本哲学。

从 linear 到 multilinear 的基本动作是用 Cauchy-Schwarz 或分解降低复杂度，同时保留主要信息。

1. Cauchy-Schwarz 的真正作用

Cauchy-Schwarz 不只是让估计变粗。好的使用方式是：降低复杂度，同时几乎不损失主要信息。例如

$$\left|\sum_n a_n b_n\right|^2\le \left(\sum_n|a_n|^2\right)\left(\sum_n|b_n|^2\right).$$

如果右边的两个平方和更容易理解，这一步就是有效的。

2. Fourier 分解

在调和分析中，一个函数被切成频率块：

$$f=\sum_\theta f_\theta.$$

线性估计可能无法直接控制 $\sum_\theta f_\theta$，但双线性或多线性估计能利用不同 $\theta$ 之间的 transversal 结构。

3. Multiscale 理论

Kakeya 和 restriction 中，多尺度分解把一个集合或函数切成不同尺度上的 pieces。每一层可能只给出局部信息，但层与层之间的递归关系能产生全局维数或范数估计。

4. BSZ 准则

Bourgain-Sarnak-Ziegler 准则就是线性到双线性的典型例子。要证明

$$\sum_{n\le N}\mu(n)a_n=o(N),$$

可以转而控制

$$\sum_{n\le N}a_{pn}\overline{a_{qn}}$$

对不同素数 $p\ne q$ 的相关。这把乘法函数的线性相关转成序列自身的双线性相关。

5. 方法的边界

多线性化不是免费午餐。右边的量可能更复杂，分解后还要重新 glue 回整体估计。好的多线性方法，总是在“降低复杂度”和“保持信息”之间找到平衡。

2017年10月30日