polynomials and algebraic operations












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I am trying to study algebraic properties of solutions of differential equations and hoping to get some guidance from the point of view of algebraic geometry.



Let $frac{du^k}{dt}=F^k(u^1,...,u^n)$, $k=1,...,n$, be an autonomous system of differential equations, where $F^k: mathbb{C^n}rightarrow mathbb{C^n} $ are sufficiently smooth functions that may have only isolated zeros and infinities.



Let
$u^k=u^kleft(t,C^1,C^2,...,C^nright)$, $k=1,...,n$, be a general solution of the differential system above, where $C^1,...,C^n$ are constants of integration,
and $phi^1(u)=t+C^1$, $phi^k(u)=C^k$, $k=2,...,n$.



If $u_1, u_2$ are two solutions, let us define a binary operation $*$ on the set of solutions as follows: $phi^k(u_1*u_2)=ln{left[e^{phi^k(u_1)}+e^{phi^k(u_2)}right]}+Q^kleft(phi^1(u_2)-phi^1(u_1),...,phi^n(u_2)-phi^n(u_1)right)$, where $Q^k$ is a polynomial.



Originally $Q^k$ is an arbitrary function, but here I am assuming if it is a polynomial in $k$ variables over $mathbb{C}$ this will attract attention of algebraic geometers (or at least I hope it will).



I found that fixed solutions (or stationary points) of the system form a closed set under this operation. Same can be said about periodic solutions.



Also, if $Q$ is even, then the operation $*$ is commutative, and if $Q=0$, it is associative.



I want to find more properties of the operation $*$.






differential-equations algebraic-geometry







share|cite|improve this question





























    1














    I am trying to study algebraic properties of solutions of differential equations and hoping to get some guidance from the point of view of algebraic geometry.



    Let $frac{du^k}{dt}=F^k(u^1,...,u^n)$, $k=1,...,n$, be an autonomous system of differential equations, where $F^k: mathbb{C^n}rightarrow mathbb{C^n} $ are sufficiently smooth functions that may have only isolated zeros and infinities.



    Let
    $u^k=u^kleft(t,C^1,C^2,...,C^nright)$, $k=1,...,n$, be a general solution of the differential system above, where $C^1,...,C^n$ are constants of integration,
    and $phi^1(u)=t+C^1$, $phi^k(u)=C^k$, $k=2,...,n$.



    If $u_1, u_2$ are two solutions, let us define a binary operation $*$ on the set of solutions as follows: $phi^k(u_1*u_2)=ln{left[e^{phi^k(u_1)}+e^{phi^k(u_2)}right]}+Q^kleft(phi^1(u_2)-phi^1(u_1),...,phi^n(u_2)-phi^n(u_1)right)$, where $Q^k$ is a polynomial.



    Originally $Q^k$ is an arbitrary function, but here I am assuming if it is a polynomial in $k$ variables over $mathbb{C}$ this will attract attention of algebraic geometers (or at least I hope it will).



    I found that fixed solutions (or stationary points) of the system form a closed set under this operation. Same can be said about periodic solutions.



    Also, if $Q$ is even, then the operation $*$ is commutative, and if $Q=0$, it is associative.



    I want to find more properties of the operation $*$.






    differential-equations algebraic-geometry







    share|cite|improve this question



























      1












      1








      1


      1





      I am trying to study algebraic properties of solutions of differential equations and hoping to get some guidance from the point of view of algebraic geometry.



      Let $frac{du^k}{dt}=F^k(u^1,...,u^n)$, $k=1,...,n$, be an autonomous system of differential equations, where $F^k: mathbb{C^n}rightarrow mathbb{C^n} $ are sufficiently smooth functions that may have only isolated zeros and infinities.



      Let
      $u^k=u^kleft(t,C^1,C^2,...,C^nright)$, $k=1,...,n$, be a general solution of the differential system above, where $C^1,...,C^n$ are constants of integration,
      and $phi^1(u)=t+C^1$, $phi^k(u)=C^k$, $k=2,...,n$.



      If $u_1, u_2$ are two solutions, let us define a binary operation $*$ on the set of solutions as follows: $phi^k(u_1*u_2)=ln{left[e^{phi^k(u_1)}+e^{phi^k(u_2)}right]}+Q^kleft(phi^1(u_2)-phi^1(u_1),...,phi^n(u_2)-phi^n(u_1)right)$, where $Q^k$ is a polynomial.



      Originally $Q^k$ is an arbitrary function, but here I am assuming if it is a polynomial in $k$ variables over $mathbb{C}$ this will attract attention of algebraic geometers (or at least I hope it will).



      I found that fixed solutions (or stationary points) of the system form a closed set under this operation. Same can be said about periodic solutions.



      Also, if $Q$ is even, then the operation $*$ is commutative, and if $Q=0$, it is associative.



      I want to find more properties of the operation $*$.






      differential-equations algebraic-geometry







      share|cite|improve this question















      I am trying to study algebraic properties of solutions of differential equations and hoping to get some guidance from the point of view of algebraic geometry.



      Let $frac{du^k}{dt}=F^k(u^1,...,u^n)$, $k=1,...,n$, be an autonomous system of differential equations, where $F^k: mathbb{C^n}rightarrow mathbb{C^n} $ are sufficiently smooth functions that may have only isolated zeros and infinities.



      Let
      $u^k=u^kleft(t,C^1,C^2,...,C^nright)$, $k=1,...,n$, be a general solution of the differential system above, where $C^1,...,C^n$ are constants of integration,
      and $phi^1(u)=t+C^1$, $phi^k(u)=C^k$, $k=2,...,n$.



      If $u_1, u_2$ are two solutions, let us define a binary operation $*$ on the set of solutions as follows: $phi^k(u_1*u_2)=ln{left[e^{phi^k(u_1)}+e^{phi^k(u_2)}right]}+Q^kleft(phi^1(u_2)-phi^1(u_1),...,phi^n(u_2)-phi^n(u_1)right)$, where $Q^k$ is a polynomial.



      Originally $Q^k$ is an arbitrary function, but here I am assuming if it is a polynomial in $k$ variables over $mathbb{C}$ this will attract attention of algebraic geometers (or at least I hope it will).



      I found that fixed solutions (or stationary points) of the system form a closed set under this operation. Same can be said about periodic solutions.



      Also, if $Q$ is even, then the operation $*$ is commutative, and if $Q=0$, it is associative.



      I want to find more properties of the operation $*$.






      differential-equations algebraic-geometry




      differential-equations algebraic-geometry






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      edited yesterday

























      asked Jan 3 at 21:34









      Zviad Khukhunashvili

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