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There are nine coefficients in an equation of cubic curve. It means nine points can determine a unique cubic curve. But two cubic curves A, B can have nine intersection points. If we use the nine intersection points to draw a cubic curve, then why do we get at least two cubic curves,not a unique one?

jscoot
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    What's your definition of a cubic curve? Is it only two dimensions? Can you have any term with degree at most 3? – T.J. Gaffney Dec 02 '19 at 06:20
  • There are ten coefficients, though. But because multiplying all coefficients by the same non-zero constant, the space of cubic curves is indeed nine-dimensional. – Arthur Dec 02 '19 at 06:38
  • Do two cubic curves have nine points of intersection? Bezout's theorem requires projective spaces, so I guess there's a point at infinity which accounts for one of the intersection points (since generally cubics are unbounded). Are you using a different theorem? – T.J. Gaffney Dec 02 '19 at 06:41
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    Here is an idea for a resolution to your conundrum: All cubic curves that go through eight of the points will go through the ninth. So if you are unlucky with how you choose your points (such as choosing them based on where it intersects another cubic), one of the points will be redundant. Is this true? I don't know. – Arthur Dec 02 '19 at 06:44
  • Then why are we sure eight points can't be redundant? @Arthur – jscoot Dec 02 '19 at 10:53
  • Two cubic curves can have nine points of intersection. That is for sure because by solving the algebraic equations in third degree we can find nine solutions. Each cubic curve can intersect with the line at infinity at three different points. But two cubic curves can intersect with the line at infinity at different points just like two non-parallel lines intersect at finite points but only two parallel lines intersect at the point on the line at infinity. @T.J. Gaffney – jscoot Dec 02 '19 at 10:56

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I came here with the same question. Seeing that there is still no answer, I played around in Mathematica and this is my understanding so far.

There are 10 coefficients in an equation for a cubic. Let us first consider the general case. In a general case, all coefficients are non-zero, and we can normalize the constant to be, say, 1. Then 9 points give us a system of linear equations with rank 9. It will have a unique solution defining a unique cubic curve.

If we choose our 9 points as the intersection points of two distinct cubic curves, we are no longer in the general case. Here we have a system of linear equations that admits two different solutions, which can only happen if the rank is less than 9. Indeed, if I try it in Mathematica, I get a system with rank 8. Of course, then there are infinitely many solutions. Or, in other words, we can take an arbitrary 10th point, and draw a curve that passes through the initial 9 points plus the new point.

Andrei
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