Question about handling algebraic root to Minimal polynomial

Randall · Sat May 15, 2010 3:48 am

Can Singular find the minimal polynomial and its degree for the algebraic number = 0.52826258796915823872496738067794327575644009724406207212291796494951276379419026203688924917511315575571328699287880446279776976305813835739736940046222158067720539130899894885367006337137020013305147010199123617961585114205821143595991652324421491994272816876555253994851554740221902434937009202886513118246919636919640408207581115749837261188909068239824331152491070586432440739764377554324452942774186344294013065344037046705935128385500285778302968147258523201223587876055498459161608368840282542295632408237368443946534508327451003611890728345444954824293710596173806620847043713135206692682598861771886319327302204816797994329632257669297692314491258994096767465520962292617665186980266428239807137779956548394936517339991366780850243952740234825957967718934557481973261016648484323688616566707236330983682862279585572148366029930056475854633811288025459947865362741302493618885491934146652328993466212500471355378592490616194660877934635991571074642274565475913010863459037388651563330761761286201559025566697977898327220957386289643 ?

This is from a Gram matrix (inner products) of a set of points in R3. It is conjectured that the points are algebraic.

I would like to find the minimal polynomial and degree (if possible) While other symbolic programs offer an ability, it usually comes down to finding a linear dependence upon powers of the root (this number) up to a degree (which at the moment no one knows for this number)

Does Singular handle this with ease?

Re: Question about handling algebraic root to Minimal polynomial

steenpass · Mon May 17, 2010 10:34 am

Up to now, Singular cannot handle the computation of minimal polynomials up to a given precision. When computing minimal polynomials, all the computations are done in a symbolic manner without rounding errors.
Thus, working over Q in Singular, your number is regarded as an element of the given ring, and working over Z, the minimal polynomial is something like 10^n*x-52826 which is not what you mean, I guess.

If you or someone else is interested in implementing the algorithms you need in Singular, the Singular developer team could give you further assistance on how to do this.

Regards,
Andreas

Re: Question about handling algebraic root to Minimal polynomial

gorzel · Wed May 19, 2010 5:16 pm

What you have in mind, is to see whether this finite decimal expansion correspond to
an algebraic number.

There is Simon Plouffe's http://pictor.math.uqam.ca/~plouffe/
old
Inverse Symbolic calculator:
http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html

which deals wtih these questons to recognize a special number from its first terms.

Entering your number, it give the following results:

Code:

* Simple Lookup and Browser for any number.

 Your value of 5282625879691582 would be here.

 5282626607578793 = (0261) 2+3*x-x^2-x^3-x^4+5*x^5
 5282627652776098 = (0261) -3-2*x+4*x^2-3*x^3+6*x^4+2*x^5
 5282628176388586 = (0263) 5+5*x+16*x^3
 5282631423533001 = (0261) 2+4*x-5*x^2-3*x^3+6*x^4-x^5
 5282636481755707 = (0261) -4+5*x+4*x^2+x^4+4*x^5

*  Integer Relation Algorithms  for any number.
    K does NOT satisfy a polynomial equation
    with small coefficients of degree <=5.

As far as I see, one tool behind (apart from the lookuptables) is the LLL-Algorithm
to find the integral coefficients for some minimal polynomial.

The LLL-algorithm is implemeted in the lll.lib, but this does not mean that
the same wizard as the ISC i available with Singular.

Since Plouffe used Maple and PARI for the ISC,
PARI-GP will be the right adress for your question.

As the above example shows, it is not so important to have so many digits, the ISC
only takes the first 16. What matters for the search is the (unknown) degree of the minpoly.

Visit also the other links at

http://en.wikipedia.org/wiki/Simon_Plouffe

to learn more about the ISC and the stories behind.

----

Ch. Gorzel