Heat flow 与多项式零点：从变形思想到 Riemann Hypothesis 的 toy model

旧博客原文

原题：Heat flow and the zero of polynomial-a approach to Riemann Hypesis

this is a note after reading the blog:Heat flow and the zero of polynomial.

1.instead of consider the original version:

$\partial_{zz}f(z,t)=\partial_tf(z,t)$ .

consider the corresponding “equidistribution version” is also interesting:

$\partial_{zz}f(z,t)=\theta(z,t)\partial_tf(z,t)$ ,especially $\theta(z,t)=e^{2\pi i\alpha t},\alpha\in R-Q$ .

where $f(z)=z^n+a_{n-1}z^{n-1}+...+a_1z+a_0$ .

$f(z,t)=\sum_{k=1}^n\sum_{0\leq m\leq k-2,2|k-m}\frac{k!}{m!(k-m)!}z^mt^{k-m}.$

$=\sum_{k=1}^m\sum_{0\leq m\leq k-2,2|k-m}C_k^mt^{k-m})z^mt^{k-m}$

$\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_k^mt^{k-m})z^m$ .

rescaling:

$F_t:(z_1(t),...,z_n(t))\longrightarrow (\frac{z_1(t)}{t},...,\frac{z_n(t)}{t})$ .

$F_t\cdot f(z,t)=\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_{k}^mt^{k-n})z^m$ .

$\lim_{t\to \infty}F_t\cdot f(z,t)=\sum_{m=0,2|n-m}^{n-2}C_n^mz^m$ .(*)

even term $\longrightarrow$ constant.(after renormelization)

odd term $\longrightarrow$ 0(invariant).so at least the sum zeros of is invarient.

by the algebraic fundamental theorem,we have n zero $\{z_1,...,z_n\}$ of (*).

until now,we already now if the n zeros is distinct,then because the energy is the energy is the same and the entropy is increase so $\exists T>>0,\forall t_i,t_j>T$ , $\{t>T|z_i(t)\} \cap \{t>T|z_j(t)\}=\emptyset$ . $\lim_{t\to \infty}|z_i(t)|=\infty$ and $\lim_{t\to \infty}arg(z_i(t))=z_i$ .

but how to know the information of the change of direction at “blow up” time?

1.change direction only at $blow up$ .

2.energy invariant $\sum_{1\leq i\neq j\leq n}\frac{1}{|x_i-x_j|^2}$ .

3.general philosophy

deformation some function under some evolution equation, such like heat equation,wave equation,shrodinger equation.and there is some conversion thing under the equation,and some quantity that could calculate directly such like the trace of spectral.

4.difficultis

this philosophy could generate to the analytic function case,but to make the limit case(I only know how ti deal with this now)coverage.we need very good control on the coefficient.

and to investigate the change of direction at blow up point maybe we need some knowledge about the burid group.

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。

用 heat flow 研究多项式零点，是理解更复杂解析函数零点问题的一个 toy model。基本思想是：让函数随时间演化，观察零点如何移动，以及哪些量在演化中保持或单调。

Heat flow 与多项式零点：从变形思想到 Riemann Hypothesis 的 toy model — heat flow 让多项式零点随时间运动，提供研究零点实性和不变量的 toy model。

1. 多项式的 heat deformation

设 $P(x)$ 是多项式，考虑

$$\partial_t u=\partial_x^2u,\qquad u(0,x)=P(x).$$

因为 heat operator 保持多项式空间，$u(t,x)$ 仍然是多项式。其零点随 $t$ 移动。

2. 不变量与单调量

某些系数组合在演化中保持不变，另一些量具有单调性。例如最高次项不变，低阶偶次项会随 heat flow 改变。零点的质心或某些对称量可能保持。

3. 零点碰撞

若零点始终实且互异，运动图像较清楚；真正困难发生在零点碰撞或分裂时。此时需要理解 blow-up 时间附近的方向变化。

4. 与 Riemann Hypothesis 的类比

de Bruijn-Newman 常数研究的是 Xi 函数在 heat flow 型变形下零点保持实的临界时间。多项式模型不能证明 RH，但能展示同一种哲学：通过演化方程追踪零点几何。

5. 需要的估计

要从多项式推广到整函数，必须控制系数、增长阶和极限过程。多项式情形的代数基本定理给出有限零点；整函数情形需要更强的紧性和零点分布估计。

Heat flow 与多项式零点：从变形思想到 Riemann Hypothesis 的 toy model

旧博客原文

补充说明

1. 多项式的 heat deformation

2. 不变量与单调量

3. 零点碰撞

4. 与 Riemann Hypothesis 的类比

5. 需要的估计

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