Recover polynomial from 1 root

09212018, 01:17 PM
Post: #1




Recover polynomial from 1 root
I were reading Mathematical Universe, by William Dunham
On page 210, he showed how an algebraic number must be a root of a specific polynomial (with integer coefficient) A hard example, r = sqrt(6) / (5^(1/3) + sqrt(3)), is a solution of ... 4 x^12  49248 x^10  37260 x^8  127440 x^6 + 174960 x^4  139968 x^2 + 46656 = 0 How does he do that ? What is the trick to recover polynomial from a single root ? BTW, anyone who has HP Prime, can you confirm r really is a root of that polynomial ? 

09212018, 02:15 PM
Post: #2




RE: Recover polynomial from 1 root
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Hi, Albert Chan: (09212018 01:17 PM)Albert Chan Wrote: I were reading Mathematical Universe, by William Dunham Excellent book, which I bought many years ago (both in Spanish ["El Universo Matemático"] and English. Recommended. Quote:On page 210, he showed how an algebraic number must be a root of a specific polynomial (with integer coefficient). A hard example, r = sqrt(6) / (5^(1/3) + sqrt(3)), is a solution of ... Any given algebraic number such as your r is the root of an infinite number of polynomials, so the one you quoted is usually the minimal polynomial (which is unique) for that algebraic number. The minimal polynomial can be found with most advanced software (the function is usually called something like "minpol") which do use lattice reduction algorithms such as LLL, or in more recent times, PSLQ. [Of course it can also be done manually by performing some very tedious but purely algebraic manipulations (raising xr to suitable powers and isolating radicals at one side, rinse and repeat, until you get rid of all radicals).] I did include a minpol functionality in version 2.0 of my IDENTIFY program for the HP71B which does just that, find the minimal polynomial for any given input. If the number is indeed algebraic, the polynomial will have it as one of its roots, else the root will be an approximation to the nonalgebraic number given (say Pi). Quote:BTW, anyone who has HP Prime, can you confirm r really is a root of that polynomial ? You don't need a Prime for that, an HP71B will do as well, as will many other advanced HP models. Regards. V. . Find All My HPrelated Materials here: Valentin Albillo's HP Collection 

09212018, 02:38 PM
(This post was last modified: 09212018 06:46 PM by Albert Chan.)
Post: #3




RE: Recover polynomial from 1 root
Thanks, Valentin Albillo
I had noticed the polynomial is not fully reduced (top coefficient of 4 can be removed). Seems Dunham uses software to get the polynomial too 

09212018, 04:12 PM
(This post was last modified: 09212018 04:22 PM by Albert Chan.)
Post: #4




RE: Recover polynomial from 1 root
FYI, this is what manual "rinse and repeat" method look like:
x = r = sqrt(6) / (5^(1/3) + sqrt(3)) 3*sqrt(2) x + 5400^(1/6) x = 6 < multiply 6/r, both side 5400^(1/6) x = 6  3*sqrt(2) x 5400 x^6 = (6  3*sqrt(2) x)^6 < only square root remains ... Expand above, and group sqrt(2) terms, we get sqrt(2) * (81 x^5 + 540 x^3 + 324 x) = x^6 + 405 x^4 + 810 x^2 + 108 Square both side, all radicals are gone. We get the polynomial: x^12  12312 x^10  9315 x^8  31860 x^6 + 43740 x^4  34992 x^2 + 11664 = 0 

09212018, 07:48 PM
Post: #5




RE: Recover polynomial from 1 root  
10092018, 12:37 PM
(This post was last modified: 10092018 03:17 PM by Albert Chan.)
Post: #6




RE: Recover polynomial from 1 root
Using XCas simplify on root r, I were expecting little or no effect, but I get this:
r := sqrt(6) / (5^(1/3) + sqrt(3)) simplify(r) => rootof( [3018245, 37511698, ... (Total 12 numbers) ], [1, 0, 54, 20 ... (Total 13 numbers)]]) Does above have any relation to the polynomial that have r as a root ? 

10122018, 07:34 AM
Post: #7




RE: Recover polynomial from 1 root
(10092018 12:37 PM)Albert Chan Wrote: Using XCas simplify on root r, I were expecting little or no effect, but I get this: And hence you've discovered why we, after much persuasion, convinced Dr Parisse to turn "rootof" off in Prime. Nobody can understand WTH it is supposed to be describing or is useful for. Despite having heard the explanation many times it still is a mystery... TW Although I work for the HP calculator group, the views and opinions I post here are my own. 

10132018, 09:52 PM
(This post was last modified: 10132018 09:53 PM by Valentin Albillo.)
Post: #8




RE: Recover polynomial from 1 root
(10122018 07:34 AM)Tim Wessman Wrote:(10092018 12:37 PM)Albert Chan Wrote: Using XCas simplify on root r, I were expecting little or no effect, but I get this: Indeed. I do understand what it does but, frankly, it's next to useless in most cases and the result is highly unfathomable as Albert Chan has experienced first hand. I would suggest renaming this existing functionality to something that best describes what it does (if at all possible) and using instead the name "rootof" to return the minimum polynomial which has the input as a root (i.e.: the wellknown minpol functionality). That would be much more generally useful, IMHO. V. . Find All My HPrelated Materials here: Valentin Albillo's HP Collection 

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