Timeline for Real algebraic curves of $\mathbb R^2\sim\mathbb C$ containing all zeroes of infinitely many rational polynomials
Current License: CC BY-SA 2.5
9 events
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| May 3, 2010 at 13:57 | comment | added | Roland Bacher | Such curves are of course invariant with respect to complex reflection. They might however not be simple and consist,say ,of two curves lacking this property. | |
| May 1, 2010 at 13:15 | history | edited | Dror Speiser | CC BY-SA 2.5 |
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| May 1, 2010 at 12:30 | comment | added | damiano | @Dror: let C be a curve with the required property. The set of points that we are interested in is infinite and hence it is dense in some irred component of C: replace C by this component. Thus, there is a Zariski dense set of pts (x,y) with (x,-y) in C. It follows that the symmetry $y \mapsto -y$ fixes C and if f(x,y) vanishes on C, then also f(x,-y) vanishes on C. We conclude that an equation defining C must involve only even or only odd powers of y. Having said that, I also suspect that the property is quite rare! | |
| Apr 30, 2010 at 19:11 | comment | added | Dror Speiser | @damiano: I was thinking about this earlier as well. Can you prove the statement about powers of $y$? I actually believe that almost all curves will fail to have the property. | |
| Apr 30, 2010 at 16:55 | comment | added | damiano | @roland-bacher: ok, so you want examples of curves whose points contain all the zeros of infinitely many monic irreducible polynomials with rational coefficients. Dror's argument shows that an irreducible such curve must contain only even or only odd powers of y. Thus either it is the curve y=0 or it must contain only even powers of y. Is this in line with the question that you want answered? | |
| Apr 30, 2010 at 15:35 | comment | added | Dror Speiser | I don't understand. Doesn't my example lack the property? The curve you mention is different. | |
| Apr 30, 2010 at 14:57 | comment | added | Roland Bacher | The (non-simple) curve given by $x^2-y^2$ falls into the trivial class of examples mentioned in the question. I know there are many such curves. I ask for specific examples which lack the property. | |
| Apr 30, 2010 at 14:11 | history | edited | Dror Speiser | CC BY-SA 2.5 |
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| Apr 30, 2010 at 13:56 | history | answered | Dror Speiser | CC BY-SA 2.5 |