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M.G.
M.G.
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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 like a very natural. If we set $\gamma(t)=x(t)+iy(t)$, this amounts to the equation $$ \int_{\gamma_t}\frac{dz}{F(z)}=t $$ 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) $F$.

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 like a very natural. If we set $\gamma(t)=x(t)+iy(t)$, this amounts to the equation $$ \int_{\gamma_t}\frac{dz}{F(z)}=t $$ 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) $F$.

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 $$ \int_{\gamma_t}\frac{dz}{F(z)}=t $$ 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) $F$.

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M.G.
M.G.
  • 7.1k
  • 3
  • 46
  • 60

Planar polynomial vector field for a harmonic pair of polynomials

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 like a very natural. If we set $\gamma(t)=x(t)+iy(t)$, this amounts to the equation $$ \int_{\gamma_t}\frac{dz}{F(z)}=t $$ 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) $F$.