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From: Carl Woll on 30 Dec 2009 04:18 Andre Hautot wrote: > x= Sqrt[2] + Sqrt[3] + Sqrt[5] is an algebraic number > > MinimalPolynomial[Sqrt[2] + Sqrt[3] + Sqrt[5], x] > > returns the polynomial : 576 - 960 x^2 + 352 x^4 - 40 x^6 + x^8 as > expected > > Now suppose we only know the N first figures of x (N large enough), say > : N[x,50] = 5.3823323474417620387383087344468466809530954887989 > > is it possible to recognize x as a probably algebraic number and to > deduce its minimal polynomial ? > > Thanks for a hint, > ahautot > > > That's what RootApproximant is for: In[6]:= RootApproximant[5.3823323474417620387383087344468466809530954887989] Out[6]= Root[576 - 960 #1^2 + 352 #1^4 - 40 #1^6 + #1^8 &, 8] Carl Woll Wolfram Research
From: Artur on 30 Dec 2009 04:18 RootApproximant[5.3823323474417620387383087344468466809530954887989] Root[576 - 960 #1^2 + 352 #1^4 - 40 #1^6 + #1^8 &, 8] If you have older version 5 you can use inspite RootApproximant function Recognize Best wishes Artur Andre Hautot pisze: > x= Sqrt[2] + Sqrt[3] + Sqrt[5] is an algebraic number > > MinimalPolynomial[Sqrt[2] + Sqrt[3] + Sqrt[5], x] > > returns the polynomial : 576 - 960 x^2 + 352 x^4 - 40 x^6 + x^8 as > expected > > Now suppose we only know the N first figures of x (N large enough), say > : N[x,50] = 5.3823323474417620387383087344468466809530954887989 > > is it possible to recognize x as a probably algebraic number and to > deduce its minimal polynomial ? > > Thanks for a hint, > ahautot > > > >
From: dh on 30 Dec 2009 04:19 Hi, algebraic numbers are dense in R. Therefore there are an infinite number of algebraic numbers "close" to any rational (here even : finite decimal representation). Therefore, you must give a more stringent condition, to choose one. Daniel On 29 Dez., 07:24, Andre Hautot <ahau...(a)ulg.ac.be> wrote: > x= Sqrt[2] + Sqrt[3] + Sqrt[5] is an algebraic number > > MinimalPolynomial[Sqrt[2] + Sqrt[3] + Sqrt[5], x] > > returns the polynomial : 576 - 960 x^2 + 352 x^4 - 40 x^6 + x^8 as > expected > > Now suppose we only know the N first figures of x (N large enough), say > : N[x,50] = 5.3823323474417620387383087344468466809530954887989 > > is it possible to recognize x as a probably algebraic number and to > deduce its minimal polynomial ? > > Thanks for a hint, > ahautot
From: DrMajorBob on 31 Dec 2009 03:16 Strictly speaking, you're right. The following results (if they were to be minimal polynomials EXACTLY) should be different: 5.3823323474417620387383087344468466809530954887989 // RootApproximant Root[576 - 960 #1^2 + 352 #1^4 - 40 #1^6 + #1^8 &, 8] 5.38233234744176203873830873444684668095309548879891 // RootApproximant Root[576 - 960 #1^2 + 352 #1^4 - 40 #1^6 + #1^8 &, 8] This is a little like those idiotic SAT and GRE questions that ask "What's the next number in the following series?"... where any number will do. Test writers don't seem to know there's an interpolating polynomial (for instance) to fit the given series with ANY next element. Bobby On Wed, 30 Dec 2009 03:12:00 -0600, Francesco <fracix(a)gmail.com> wrote: > > "Andre Hautot" <ahautot(a)ulg.ac.be> ha scritto nel messaggio > news:hhc7a1$2o2$1(a)smc.vnet.net... >> x= Sqrt[2] + Sqrt[3] + Sqrt[5] is an algebraic number >> >> MinimalPolynomial[Sqrt[2] + Sqrt[3] + Sqrt[5], x] >> >> returns the polynomial : 576 - 960 x^2 + 352 x^4 - 40 x^6 + x^8 as >> expected >> >> Now suppose we only know the N first figures of x (N large enough), say >> : N[x,50] = 5.3823323474417620387383087344468466809530954887989 >> >> is it possible to recognize x as a probably algebraic number and to >> deduce its minimal polynomial ? > > I have the impression that in your case the MinimalPolynomial is simply > x-N. > > If you add a figure to N, let say ......9891, you will get a different > polynomial. > > > > __________ Information from ESET Smart Security, version of virus > signature database 4725 (20091229) __________ > > The message was checked by ESET Smart Security. > > http://www.eset.com > > > > > -- DrMajorBob(a)yahoo.com
From: Robert Coquereaux on 31 Dec 2009 03:17
"Impossible....Not at all" I think that one should be more precise: Assume that x algebraic, and suppose you know (only) its first 50 digits. Then consider y = x + Pi/10^100. Clearly x and y have the same first 50 digits , though y is not algebraic. Therefore you cannot recognize y as algebraic from its first 50 digits ! The quoted comment was in relation with the question first asked by hautot. Now, it is clear that, while looking for a solution x of some equation (or definite integral or...), one can use the answer obtained by applying RootApproximant (or another function based on similar algorithms) to numerical approximations of x, and then show that the suggested algebraic number indeed solves exactly the initial problem. If so, you will indeed have recognized the number x as algebraic, from its first N figures. But this does not seem to be the question first asked by hautot. Also, if one is able to obtain information, for any N, on the first N digits of a real number x, this is a different story... and a different question. Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit : > >> To recognize a number x as algebraic, from its N first figures, is >> impossible. > > Not at all. There are polynomial factorization algorithms based on > this notion (maybe you knew that). > > Daniel Lichtblau > Wolfram Research |