In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is naturally arise from a polynomial vector field?

More precisely, let $X=P\partial_x+Q\partial_y$ be a polynomial vector field. This generates a differential operator $L$ with $L(u)=Pu_x+Qu_y$. Is there always a Green's function for $L$ defined on the complement of singularities of $X$? Can it be represented as a usuall function, not merely as a distribution? Can one imagine a possible relation between the dynamical behavior of $X$ and some properties of the corresponding Green functions?


1 Answer 1


This is not a answer,but some comments.At first,I want to say the Hilbert 16th problem is very difficult,and there is a lot of way to approach the problem.when I learn the ordinary differential equation 3 years ago,I consider the problem merely in a topological way.

But there I image you want to investigate the perbutation of the vector field under certain PDE,such as heat flow,wave flow.

Let $P(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ be a monic polynomial of degree $n$ with complex coefficients. Then by the fundamental theorem of algebra, we can factor $P$ as

$\displaystyle P(z) = (z-z_1) \dots (z-z_n) \ \ \ \ \ (1)$

for some complex zeroes $z_1,\dots,z_n$ (possibly with repetition).

Now suppose we evolve ${P}$ with respect to time by heat flow, creating a function ${P(t,z)}$ of two variables with given initial data ${P(0,z) = P(z)}$ for which

$\displaystyle \partial_t P(t,z) = \partial_{zz} P(t,z). \ \ \ \ \ (2)$

On the space of polynomials of degree at most ${n}$, the operator ${\partial_{zz}}$ is nilpotent, and one can solve this equation explicitly both forwards and I think the limit case of the dynamic system is under control,I just mean we due to we can calculate the equation $f(z,t)=\sum_{k=1}^n\sum_{0\leq m\leq k-2,2|k-m}\frac{k!}{m!(k-m)!}z^mt^{k-m}$.

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


exactly,so we can rescaling it under $F_t:f(z,t)\to f(\frac{z}{t},t)$ and take $t\to \infty$,the limit case is the equation:$\lim_{t\to \infty}F_t\cdot f(z,t)=\sum_{m=0,2|n-m}^{n-2}C_n^mz^m$(*) this have $n$ zero on $C$.

Now,let $X(t)=P(t,x)\partial x+Q(t,y)\partial y$ be a one parameter group generate by heat flow of polynomial vector field.Obeserve that the number of limit cycles does not change at the time if the roots of the polynomial do not colliade.so we change the original problem to investigate the case $t\to \infty$ and the slice moments when roots of $P(z,t)$ or $Q(z,t)$ collide along the flow.

until now,at least for the case the $n$ zero is distinct in (*),we know at last the zero will go to infinity along each direction come form the zeros of (*),so the only complicated thing is the finite “blow up”time i.e. the time zeroes must collide.these will lead to to breaken of symmetry(just think about the example $z^2=c$,there is only one equation,but two different pictures).to investigate this I think we need some knowledge about buried group.

Anyway,one key obeserve is that the points should diverging along $n$ distinct line to infinity in the complex plane $C$.at least in the case the limit polynomial has $n$ distinct root.for the degenerate situation the thing is not so easy to control we need to analysis to certain entanglement pairs.and if $deg(f)$ is odd,then we will have a root $0$ in the limit case,this is also a annoying thing. in my opinion,i think we need some knowledge from buried group to investigate a fix dynamic system generat by the zero of polynomial under heat flow,just think with the situation for quadratic polynomial $z^2+c=0$,this will corresponding to $2$ different dynamic picture,because we do not know what happen at $(0,0)$.this is a broken of symmetry.we need to use braid group and the jordan curve theorem(the topology only change when two point collide) to find the loss information.by the way,i think consider the deformation of zeros for polynomial function under heat equation on compact complex analytic surface(especially on $\hat C$) is also interesting.


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