Mixing
When do two plane cubic curves have 9 real intersection?
$begingroup$
What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an example with 9 real intersections but a characterization.
algebraic-geometry algebraic-curves real-algebraic-geometry
asked Dec 15 '18 at 14:11
quantumquantum
528210
$endgroup$
add a comment |
$begingroup$
What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an example with 9 real intersections but a characterization.
algebraic-geometry algebraic-curves real-algebraic-geometry
asked Dec 15 '18 at 14:11
quantumquantum
528210
$endgroup$
1
$begingroup$
You can only ask for $8$ since complex intersections comes in pair. But I'm not sure there is a simple criterion.
$endgroup$
– Nicolas Hemelsoet
Dec 17 '18 at 21:57
$begingroup$
you are right. I could at least give necessary conditions , e.g. they should be both M-curves.
$endgroup$
– quantum
Dec 29 '18 at 13:54
$begingroup$
Are you sure ? It doesn't seems impossible to me that two plane cubic have maximal intersections numbers without being $M$-curves (but I didn't think about it during very long).
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 15:51
$begingroup$
I'm not sure but I find it intuitive. If there is no oval for one curve, then I can imagine at most 6 points of intersection with the other curve.
$endgroup$
– quantum
Dec 29 '18 at 20:35
$begingroup$
I'm not sure, for example imagine that the oval becomes closer and closer to the other component, then when they will mix that might still be the case that there is 9 intersection points. I can try to think a bit about it that's already an interesting question.
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 20:39
add a comment |
$begingroup$
What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an example with 9 real intersections but a characterization.
algebraic-geometry algebraic-curves real-algebraic-geometry
asked Dec 15 '18 at 14:11
quantumquantum
528210
$endgroup$
What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an example with 9 real intersections but a characterization.
algebraic-geometry algebraic-curves real-algebraic-geometry
algebraic-geometry algebraic-curves real-algebraic-geometry
asked Dec 15 '18 at 14:11
quantumquantum
528210
asked Dec 15 '18 at 14:11
quantumquantum
528210
asked Dec 15 '18 at 14:11
quantumquantum
528210
asked Dec 15 '18 at 14:11
quantumquantum
528210
asked Dec 15 '18 at 14:11
quantumquantum
528210
528210
1
$begingroup$
You can only ask for $8$ since complex intersections comes in pair. But I'm not sure there is a simple criterion.
$endgroup$
– Nicolas Hemelsoet
Dec 17 '18 at 21:57
$begingroup$
you are right. I could at least give necessary conditions , e.g. they should be both M-curves.
$endgroup$
– quantum
Dec 29 '18 at 13:54
$begingroup$
Are you sure ? It doesn't seems impossible to me that two plane cubic have maximal intersections numbers without being $M$-curves (but I didn't think about it during very long).
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 15:51
$begingroup$
I'm not sure but I find it intuitive. If there is no oval for one curve, then I can imagine at most 6 points of intersection with the other curve.
$endgroup$
– quantum
Dec 29 '18 at 20:35
$begingroup$
I'm not sure, for example imagine that the oval becomes closer and closer to the other component, then when they will mix that might still be the case that there is 9 intersection points. I can try to think a bit about it that's already an interesting question.
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 20:39
add a comment |
1
$begingroup$
You can only ask for $8$ since complex intersections comes in pair. But I'm not sure there is a simple criterion.
$endgroup$
– Nicolas Hemelsoet
Dec 17 '18 at 21:57
$begingroup$
you are right. I could at least give necessary conditions , e.g. they should be both M-curves.
$endgroup$
– quantum
Dec 29 '18 at 13:54
$begingroup$
Are you sure ? It doesn't seems impossible to me that two plane cubic have maximal intersections numbers without being $M$-curves (but I didn't think about it during very long).
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 15:51
$begingroup$
I'm not sure but I find it intuitive. If there is no oval for one curve, then I can imagine at most 6 points of intersection with the other curve.
$endgroup$
– quantum
Dec 29 '18 at 20:35
$begingroup$
I'm not sure, for example imagine that the oval becomes closer and closer to the other component, then when they will mix that might still be the case that there is 9 intersection points. I can try to think a bit about it that's already an interesting question.
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 20:39
1
1
$begingroup$
You can only ask for $8$ since complex intersections comes in pair. But I'm not sure there is a simple criterion.
$endgroup$
– Nicolas Hemelsoet
Dec 17 '18 at 21:57
$begingroup$
You can only ask for $8$ since complex intersections comes in pair. But I'm not sure there is a simple criterion.
$endgroup$
– Nicolas Hemelsoet
Dec 17 '18 at 21:57
$begingroup$
you are right. I could at least give necessary conditions , e.g. they should be both M-curves.
$endgroup$
– quantum
Dec 29 '18 at 13:54
$begingroup$
you are right. I could at least give necessary conditions , e.g. they should be both M-curves.
$endgroup$
– quantum
Dec 29 '18 at 13:54
$begingroup$
Are you sure ? It doesn't seems impossible to me that two plane cubic have maximal intersections numbers without being $M$-curves (but I didn't think about it during very long).
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 15:51
$begingroup$
Are you sure ? It doesn't seems impossible to me that two plane cubic have maximal intersections numbers without being $M$-curves (but I didn't think about it during very long).
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 15:51
$begingroup$
I'm not sure but I find it intuitive. If there is no oval for one curve, then I can imagine at most 6 points of intersection with the other curve.
$endgroup$
– quantum
Dec 29 '18 at 20:35
$begingroup$
I'm not sure but I find it intuitive. If there is no oval for one curve, then I can imagine at most 6 points of intersection with the other curve.
$endgroup$
– quantum
Dec 29 '18 at 20:35
$begingroup$
I'm not sure, for example imagine that the oval becomes closer and closer to the other component, then when they will mix that might still be the case that there is 9 intersection points. I can try to think a bit about it that's already an interesting question.
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 20:39
$begingroup$
I'm not sure, for example imagine that the oval becomes closer and closer to the other component, then when they will mix that might still be the case that there is 9 intersection points. I can try to think a bit about it that's already an interesting question.
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 20:39
add a comment |
1 Answer
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This "answer" cannot be posted as a comment because it's a bit detailed (though it does not fully answer my question). I just want to say that one of the curves being an M-curve (two connected component in $mathbb P^2(mathbb R)$) is not an necessary condiiton!
This is what I did:
I took 8 random real points in the affine plane namely:
[0,1],[1,2],[-1,1],[-3,-1],[5,-1],[0,0],[5,0],[-7,3]
I solved for polynomials in two variables and total degree 3 that define a curve that passes through these points:
$$x^3(-233/5250a-33/350b)+x^2y(313/1050a+23/70b)+x^2(1331/5250a+181/350b)+xy^2(807/875a+246/175b)+xy(-883/5250a-83/350b)+x(-83/525a-8/35b)+y^3(-a-b)+y^2a+yb$$
where $a,b$ are any number.I tweaked $a,b$ so that I get two curves that are not M-curves. Here is an example
the first curve is when $a=1,b=-10$ and the other curve is when $a=10,b=-10$. The part of the curves on top intersect 4 times and the remaining parts intersect 5 times. I hope this is clear.. Nevertheless, it does not answer my question. It just tells me that it is not necessary to be an M-curve to get maximum intersections.
answered Jan 8 at 11:11
quantumquantum
528210
$endgroup$
add a comment |
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$begingroup$
This "answer" cannot be posted as a comment because it's a bit detailed (though it does not fully answer my question). I just want to say that one of the curves being an M-curve (two connected component in $mathbb P^2(mathbb R)$) is not an necessary condiiton!
This is what I did:
I took 8 random real points in the affine plane namely:
[0,1],[1,2],[-1,1],[-3,-1],[5,-1],[0,0],[5,0],[-7,3]
I solved for polynomials in two variables and total degree 3 that define a curve that passes through these points:
$$x^3(-233/5250a-33/350b)+x^2y(313/1050a+23/70b)+x^2(1331/5250a+181/350b)+xy^2(807/875a+246/175b)+xy(-883/5250a-83/350b)+x(-83/525a-8/35b)+y^3(-a-b)+y^2a+yb$$
where $a,b$ are any number.I tweaked $a,b$ so that I get two curves that are not M-curves. Here is an example
the first curve is when $a=1,b=-10$ and the other curve is when $a=10,b=-10$. The part of the curves on top intersect 4 times and the remaining parts intersect 5 times. I hope this is clear.. Nevertheless, it does not answer my question. It just tells me that it is not necessary to be an M-curve to get maximum intersections.
answered Jan 8 at 11:11
quantumquantum
528210
$endgroup$
add a comment |
$begingroup$
This "answer" cannot be posted as a comment because it's a bit detailed (though it does not fully answer my question). I just want to say that one of the curves being an M-curve (two connected component in $mathbb P^2(mathbb R)$) is not an necessary condiiton!
This is what I did:
I took 8 random real points in the affine plane namely:
[0,1],[1,2],[-1,1],[-3,-1],[5,-1],[0,0],[5,0],[-7,3]
I solved for polynomials in two variables and total degree 3 that define a curve that passes through these points:
$$x^3(-233/5250a-33/350b)+x^2y(313/1050a+23/70b)+x^2(1331/5250a+181/350b)+xy^2(807/875a+246/175b)+xy(-883/5250a-83/350b)+x(-83/525a-8/35b)+y^3(-a-b)+y^2a+yb$$
where $a,b$ are any number.I tweaked $a,b$ so that I get two curves that are not M-curves. Here is an example
the first curve is when $a=1,b=-10$ and the other curve is when $a=10,b=-10$. The part of the curves on top intersect 4 times and the remaining parts intersect 5 times. I hope this is clear.. Nevertheless, it does not answer my question. It just tells me that it is not necessary to be an M-curve to get maximum intersections.
answered Jan 8 at 11:11
quantumquantum
528210
$endgroup$
add a comment |
$begingroup$
This "answer" cannot be posted as a comment because it's a bit detailed (though it does not fully answer my question). I just want to say that one of the curves being an M-curve (two connected component in $mathbb P^2(mathbb R)$) is not an necessary condiiton!
This is what I did:
I took 8 random real points in the affine plane namely:
[0,1],[1,2],[-1,1],[-3,-1],[5,-1],[0,0],[5,0],[-7,3]
I solved for polynomials in two variables and total degree 3 that define a curve that passes through these points:
$$x^3(-233/5250a-33/350b)+x^2y(313/1050a+23/70b)+x^2(1331/5250a+181/350b)+xy^2(807/875a+246/175b)+xy(-883/5250a-83/350b)+x(-83/525a-8/35b)+y^3(-a-b)+y^2a+yb$$
where $a,b$ are any number.I tweaked $a,b$ so that I get two curves that are not M-curves. Here is an example
the first curve is when $a=1,b=-10$ and the other curve is when $a=10,b=-10$. The part of the curves on top intersect 4 times and the remaining parts intersect 5 times. I hope this is clear.. Nevertheless, it does not answer my question. It just tells me that it is not necessary to be an M-curve to get maximum intersections.
answered Jan 8 at 11:11
quantumquantum
528210
$endgroup$
This "answer" cannot be posted as a comment because it's a bit detailed (though it does not fully answer my question). I just want to say that one of the curves being an M-curve (two connected component in $mathbb P^2(mathbb R)$) is not an necessary condiiton!
This is what I did:
I took 8 random real points in the affine plane namely:
[0,1],[1,2],[-1,1],[-3,-1],[5,-1],[0,0],[5,0],[-7,3]
I solved for polynomials in two variables and total degree 3 that define a curve that passes through these points:
$$x^3(-233/5250a-33/350b)+x^2y(313/1050a+23/70b)+x^2(1331/5250a+181/350b)+xy^2(807/875a+246/175b)+xy(-883/5250a-83/350b)+x(-83/525a-8/35b)+y^3(-a-b)+y^2a+yb$$
where $a,b$ are any number.I tweaked $a,b$ so that I get two curves that are not M-curves. Here is an example
the first curve is when $a=1,b=-10$ and the other curve is when $a=10,b=-10$. The part of the curves on top intersect 4 times and the remaining parts intersect 5 times. I hope this is clear.. Nevertheless, it does not answer my question. It just tells me that it is not necessary to be an M-curve to get maximum intersections.
answered Jan 8 at 11:11
quantumquantum
528210
answered Jan 8 at 11:11
quantumquantum
528210
answered Jan 8 at 11:11
quantumquantum
528210
answered Jan 8 at 11:11
quantumquantum
528210
528210
add a comment |
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1
$begingroup$
You can only ask for $8$ since complex intersections comes in pair. But I'm not sure there is a simple criterion.
$endgroup$
– Nicolas Hemelsoet
Dec 17 '18 at 21:57
$begingroup$
you are right. I could at least give necessary conditions , e.g. they should be both M-curves.
$endgroup$
– quantum
Dec 29 '18 at 13:54
$begingroup$
Are you sure ? It doesn't seems impossible to me that two plane cubic have maximal intersections numbers without being $M$-curves (but I didn't think about it during very long).
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 15:51
$begingroup$
I'm not sure but I find it intuitive. If there is no oval for one curve, then I can imagine at most 6 points of intersection with the other curve.
$endgroup$
– quantum
Dec 29 '18 at 20:35
$begingroup$
I'm not sure, for example imagine that the oval becomes closer and closer to the other component, then when they will mix that might still be the case that there is 9 intersection points. I can try to think a bit about it that's already an interesting question.
$endgroup$
– Nicolas Hemelsoet
Dec 29 '18 at 20:39