Timeline for Question on determining the minimal polynomial for an algebraic quotient
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| May 14, 2010 at 19:06 | history | edited | Randall | CC BY-SA 2.5 |
added 494 characters in body
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| May 14, 2010 at 16:48 | vote | accept | Randall | ||
| May 14, 2010 at 5:18 | comment | added | Randall | typical coefficients in the polynomials I am using are around 10^33, so I think the calculation in Magma will slow considerably. | |
| May 14, 2010 at 2:10 | comment | added | Junkie | It was 500-750MB in Magma for a random short polynomial choice, taking 100-200s to find the CharacteristicPolynomial. f:=Polynomial([Random([-100..100]) : i in [0..36]]); g:=Polynomial([Random([-100..100]) : i in [0..24]]); g:=Polynomial(Reverse(Coefficients(g))); f:=PolynomialRing(Rationals())!f/LeadingCoefficient(f); g:=PolynomialRing(Rationals())!g/LeadingCoefficient(g); P:=KroneckerProduct(CompanionMatrix(f),CompanionMatrix(g)); time char_poly:=CharacteristicPolynomial(P); But if you have larger coefficients it might explode. | |
| May 14, 2010 at 1:55 | answer | added | Gerry Myerson | timeline score: 4 | |
| May 14, 2010 at 1:21 | comment | added | Randall | I followed Qiaochu's method and implemented an algorithm in GP-Pari. I took advantage of the charpoly() command to find the characteristic polynomial. Too bad the multiple precision 925x925 Kronecker product is chewing up all the system memory, so this might put a stop to finding an answer. Thanks for the solution. Randall | |
| May 13, 2010 at 18:21 | comment | added | Kevin Buzzard | That's a much better answer than mine. You should post it. | |
| May 13, 2010 at 15:53 | comment | added | Qiaochu Yuan | You can reverse the coefficients of the polynomial for A to get the polynomial for A^{-1}. After that it is straightforward to write down a matrix whose characteristic polynomial has B = C * A^{-1} as a root by taking the Kronecker product of the appropriate companion matrices (en.wikipedia.org/wiki/Companion_matrix). Good algorithms are known for computing the characteristic polynomial; see, for example, modular.math.washington.edu/books/modform/modform/… . | |
| May 13, 2010 at 15:30 | answer | added | Kevin Buzzard | timeline score: 1 | |
| May 13, 2010 at 15:22 | history | asked | Randall | CC BY-SA 2.5 |