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Math Help - A equation of a quintic curve with 6 or at least 5 double points

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    A equation of a quintic curve with 6 or at least 5 double points

    Hi

    I ask for the equation of a quintic curve with 6 or at least 5 double points!
    I have founded the equation with only 4 double points...
    Help me, please!

    Scifo from Italy
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  2. #2
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    Quote Originally Posted by scifo View Post
    Hi

    I ask for the equation of a quintic curve with 6 or at least 5 double points!
    I have founded the equation with only 4 double points...
    Help me, please!

    Scifo from Italy
    What do you mean by double points?
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    double points of F(x,y)=0 if Fx=Fy=0

    The double points of a curve F(x,y)=0 is the points where the prime
    derivative respect to x e to y is zero (but not all the seconds derivative)
    I think...it is correct?
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    Quote Originally Posted by scifo View Post
    The double points of a curve F(x,y)=0 is the points where the prime
    derivative respect to x e to y is zero (but not all the seconds derivative)
    I think...it is correct?
    A more usual definition of a double point on a curve is that it is a point where the curve crosses itself. For a more precise definition, see here. On that same Wikipedia page, there is a graph of the folium of Descartes. That is a cubic curve, with the equation x^3+y^3 - 3xy = 0, which has a double point at the origin. The unit circle x^2+y^2-1 = 0 cuts the folium of Descartes at four points. So if you multiply their equations together, you get a quintic curve (x^3+y^3 - 3xy)(x^2+y^2 - 1) = 0 whose graph is the superposition of the graphs of the folium and the circle, and therefore has five double points.

    I don't know whether a curve with a quintic equation can have six double points.
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    A quintic five double points irreducible

    Hi Opalg!

    I thank you very much for your intelligent solution
    about the five double points.
    At last I have a quintic five double points!

    Perhaps you help me again for the following problems:

    May you create a quintic five double points irreducible?

    It is no hope for a quintic six double point?
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    Quote Originally Posted by scifo View Post
    May you create a quintic five double points irreducible?

    It is no hope for a quintic six double point?
    I am not knowledgeable about algebraic geometry and I cannot answer those questions. The best information I could find on the web is a 1902 Cornell University PhD dissertation by Peter Field, which you can find here. Click on "Page 1" to get a pdf file of the first chapter, which seems to state that there are many irreducible quintic curves with six double points. The text is illustrated with hand-drawn diagrams of such curves (look at Plate I at the end of that chapter), but I could not find any explicit equations for them.

    It seems that double points are known to those in the trade as "crunodes". If you have access to a university library with old books on algebraic geometry (and an Italian university ought to have lots of those ) then look for "quintic" and "crunodes" in the index. You might get lucky.
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    Perhaps a method for to find the 6-double points quintic equation?

    Hello Opalg

    I thank you for your useful information…

    I wanted a your opinion on a method, than it has sprung to mind,
    in order to find the a 6-double points quintic equation...

    Practically, fixed in the cartesian plan 8 points, 6 double and 2 simple.
    if F(x,y) is quintic polynomial and Fx(x,y) and Fy(x,y) its derivatives,
    then for every Xi,Yi of the 6 double points I have the three equations
    F(Xi,Yi)=0, Fx(Xi,Yi)=0, Fy(Xi,Yi)=0
    and for every Xj,Yj of the 2 simple points I have the equation
    F(Xj,Yj)=0.

    I have so a linear omogeneous system of 20 equations in the 20 unknown quantities
    that are the essential coefficients of the quintic, resolved which I have the equation…

    That you think some?
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  8. #8
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    Quote Originally Posted by scifo View Post
    I wanted a your opinion on a method, than it has sprung to mind,
    in order to find the a 6-double points quintic equation...

    Practically, fixed in the cartesian plan 8 points, 6 double and 2 simple.
    if F(x,y) is quintic polynomial and Fx(x,y) and Fy(x,y) its derivatives,
    then for every Xi,Yi of the 6 double points I have the three equations
    F(Xi,Yi)=0, Fx(Xi,Yi)=0, Fy(Xi,Yi)=0
    and for every Xj,Yj of the 2 simple points I have the equation
    F(Xj,Yj)=0.

    I have so a linear homogeneous system of 20 equations in the 20 unknown quantities
    that are the essential coefficients of the quintic, resolved which I have the equation…
    That sounds fairly convincing to me: 20 equations for 20 unknowns should in general have a unique solution.

    Sorry I don't have time to think about it more carefully now. I'm about to leave for a week's walking holiday in Scotland, away from the internet.
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