Timeline for Algebraic square root question
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 17, 2010 at 19:16 | comment | added | Randall | I finally was able to recover all the algebraic points, but several polynomial equations of degree 1728 (and containing coefficients of around 10^700) had to be factored. The longest time was spent taking the determinant of the resultant matrix. The highest power that occurred in the answer was 144. | |
| May 15, 2010 at 6:06 | comment | added | Junkie | It depends on the implementation, but the Kronecker-product matrices should be sparse, so the dimension is not always a worry. | |
| May 14, 2010 at 22:04 | comment | added | Randall | I actually took the time to implement the resultant method, see en.wikipedia.org/wiki/Resultant on this, particularly their paragraph under the heading Applications which states: If x and y are algebraic numbers such that P(x) = Q(y) = 0 (with degree of Q=n), we see that z = x + y is a root of the resultant (in x) of P(x) and Q(z − x) and that t = xy is a root of the resultant of P(x) and xnQ(t / x) ; combined with the fact that 1 / y is a root of ynQ(1 / y), this shows that the set of algebraic numbers is a field. It is much easy to work with matrices size m+n than size m*n. | |
| May 14, 2010 at 21:57 | comment | added | Qiaochu Yuan | Sorry, that was ambiguous. The first I is of dimension deg Q and the second I is of dimension deg P. Anyway, maybe you should just use Wiedemann's algorithm (modular.math.washington.edu/books/modform/modform/…) on the action of r+s on Q[r, s] directly instead. | |
| May 14, 2010 at 21:50 | comment | added | Pádraig Ó Conbhuí | Can you not just give the smaller polynomial 0 coefficients to make the matrices the same size? | |
| May 14, 2010 at 21:15 | comment | added | Randall | I am having problems with Lemma 4, in general the Kronecker product of A,I is not the same size matrix as I,B, so I cannot add the matrices. Furthermore, working with the Kronecker product of 864th degree polynomials is proving impractical in GP-Pari, due to both time and memory constraints, so is there a resultant method which might work better? I do need to keep the matrices with multiple precision entries down to a reasonable size. | |
| May 14, 2010 at 19:23 | vote | accept | Randall | ||
| May 14, 2010 at 19:23 | comment | added | Randall | Considering that I am working with 864 degree polynomials, and that there doesn't appear to be an easy way of implementing Lemma 2 in GP-Pari, this will take awhile, but I accept your answer. | |
| May 14, 2010 at 19:16 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
added 26 characters in body
|
| May 14, 2010 at 19:10 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |