Timeline for When do multiple polynomials have a common root?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| S Apr 22, 2022 at 23:33 | vote | accept | Niareh | ||
| S Apr 22, 2022 at 23:31 | vote | accept | Niareh | ||
| S Apr 22, 2022 at 23:33 | |||||
| Apr 22, 2022 at 23:27 | comment | added | Niareh | @Max can't you plug them into each other? The value of x can be plugged into the values for x^i? That should give $n-1$ conditions. | |
| Apr 22, 2022 at 23:26 | vote | accept | Niareh | ||
| S Apr 22, 2022 at 23:31 | |||||
| Apr 22, 2022 at 20:52 | history | edited | Konstantinos Kanakoglou |
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| Apr 22, 2022 at 2:48 | answer | added | Hunter Spink | timeline score: 6 | |
| Apr 21, 2022 at 23:47 | comment | added | Max Alekseyev | The determinant condition in EDIT is neccesary but not sufficient. The values for $x^i$ obtained from the linear system may not form powers of the same number. | |
| Apr 21, 2022 at 23:25 | comment | added | Konstantinos Kanakoglou | @Jürgen Böhm, nice! i think this generalises my answer posted below, no ? | |
| Apr 21, 2022 at 23:09 | comment | added | Jürgen Böhm | If all three polynomials are monic and their coefficients arbitrary indeterminates $a_1,\ldots,a_m, b_1,\ldots,b_n,c_1,\ldots,c_l$ you can just compute a gröbner base of $f,g,h$ in an elimination-termorder where $x > a_1,\ldots,c_l$. The elements of the gröbner base which are free of $x$ describe the values of $a_1,\ldots,c_l$ above which a common zero $x$ lies. | |
| Apr 21, 2022 at 22:57 | answer | added | Konstantinos Kanakoglou | timeline score: 1 | |
| Apr 21, 2022 at 19:00 | history | edited | Niareh | CC BY-SA 4.0 |
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| Apr 21, 2022 at 18:50 | comment | added | Niareh | There is no formula for a GCD though no? I was looking for a situation where you can write down an ideal in the coefficients of the 3 polynomials and the ideal vanishes precisely on those points where those 3 polynomials share a common root. @Derek your example seems mysterious to me even for two polynomials. But we do know the resultant works there. | |
| Apr 21, 2022 at 10:34 | comment | added | Derek | I think this may be complicated. For instance, if we take f(x) = $1+x+x^2+...x^(n_1-1)$, g(x) = $1+x+x^2+...x^(n_2-1)$, and h(x) = $1+x+x^2+...x^(n_3-1)$, it seems that there is a common factor if and only if the n's have a common divisor larger than 1. What would be the equivalent condition on the coefficients in this case? | |
| Apr 21, 2022 at 7:23 | comment | added | Somnium | Their GCD is not a constant, if they have common root. | |
| Apr 21, 2022 at 6:11 | history | asked | Niareh | CC BY-SA 4.0 |