Timeline for Find all possible rational values of the parameter of a parametric cubic such that it is reducible
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2019 at 1:25 | vote | accept | Lorenz H Menke | ||
| Mar 21, 2015 at 0:54 | comment | added | Lorenz H Menke | Yes there are two such polynomials. My notation was not very clear. Take all upper or all lower signs for the two cases. The above solution is essentially correct with a correction that the resulting polynomial in x is the form $a {x}^3 + b {x}^2 + c x$. This factors as x times a quadratic. Solving the quadratic results in a quartic discriminant. This converts to an elliptic curve with a couple of torsion points. There is one generator point hence an infinite number of solutions which is what I was looking for. Thank you all for the suggestions on the original algebraic manipulation. | |
| Mar 19, 2015 at 0:46 | comment | added | j.c. | I agree; that's how I read the question as well. | |
| Mar 18, 2015 at 12:46 | comment | added | Felipe Voloch | @j.c. But then wouldn't changing $\eta$ to $-\eta$ switch between the two? | |
| Mar 18, 2015 at 2:17 | comment | added | j.c. | I believe that there are two different polynomials under consideration, one where you always take the upper signs so that $\beta_\pm=\beta_+$ and $\beta_\mp=\beta_-$, and similarly, another polynomial corresponding to the lower signs. | |
| Mar 17, 2015 at 17:19 | comment | added | Felipe Voloch | But what does it actually mean? I would understand $\beta_+ = 1 + \eta, \beta_- = 1 - \eta$ but I don't actually know the difference between $1 \pm \eta$ and $1 \mp \eta$. | |
| Mar 17, 2015 at 17:17 | comment | added | Felipe Voloch | @j.c. Oh, I missed that, there are two different betas in the polynomial. I need to redo this. Thanks. | |
| Mar 17, 2015 at 4:38 | comment | added | j.c. | Unless I'm missing something, you get a different polynomial if you replace both $\beta_\pm$ and $\beta_\mp$ by the same variable $x$. | |
| Mar 17, 2015 at 1:52 | history | answered | Felipe Voloch | CC BY-SA 3.0 |