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During my study of plane algebraic curves, I got curious if there is a nontrivial example of a plane algebraic curve that has a node, a cusp (for my purposes I do not care which of the two kinds of cusps would the example exhibit), a tacnode, and an isolated point. By "nontrivial" I mean a curve that was not constructed as a chimera of two or more simpler curves, e.g. $(x-y)(x^2+y^2-1)=0$. Of course, it would be a quintic at the very least (i.e. the algebraic degree should be 5 at the minimum).

Apart from an explicit example, I would also be interested in a general procedure for constructing algebraic curves with a prescribed number and type of singular points.


After trying out Qiaochu's and T..'s suggestions, I have a follow-up question: does the problem become more difficult if the requirement that the curve be bounded (i.e. one can draw a circle such that the whole curve, including the isolated point, is within the circle) is imposed?

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    +1 This is a nice question. And also, I'm still laughing at the chimera thing.2010-10-23
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    If you carried out Qiaochu's and T..'s constructions and found explicit expressions for the curves, could you post them here to satisfy the curiosity of the rest of us? :)2010-10-24
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    @Rahul: Well, I've been getting curves with unbounded branches with their prescriptions... I'm still tweaking things and looking for an aesthetically pleasing arrangement of the double points, but rest assured I'll post equations and plots when I see something I like. :)2010-10-24
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    : boundedness recipe now added. I think it could sharply escalate the degree of the polynomial if you want an implicit representation P(x,y)=0.2010-10-25
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    Thanks T, I will try fixing my implementation with your suggestion.2010-10-25

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