zeros -- List the zeros of a polynomial

Synopsis

Usage:

zeros p

zeros(F, p)
Inputs:
- p, a polynomial
- F, the coefficient field; if not specified then it is taken to be the field of definition of p
Outputs:
- a list of zeros of p in the coefficient field

Description

Get a list of zeros of a polynomial using factor, baby version of rationalPoints.

i1 : QQ[x]; p = (x-2)^2 * (x^2-2) * (x^3-x-1); zeros p

o3 = {2}

o3 : List

i4 : F = toField(QQ[q]/(q^2-2)); zeros_F p

o5 = {2, q, -q}

o5 : List

i6 : F = splittingField p; #zeros_F p

o7 = 6

Note that when the degree is big, the expression of each zero in terms of a primitive element is usually complicated.

i8 : last zeros_F p

        556782 11   2772624 10   5757204 9   6650769 8   38495594 7  
o8 = - -------a   + -------a   - -------a  + -------a  - --------a  +
       8156225      8156225      8156225     8156225      8156225    
     ------------------------------------------------------------------------
     102655141 6   12461122 5   29276209 4   25076602 3   32491399 2  
     ---------a  - --------a  - --------a  - --------a  + --------a  +
      8156225       1631245      8156225      8156225      8156225    
     ------------------------------------------------------------------------
      776262    53527
     -------a + -----
     1165175    39025

o8 : F

i9 : (map(F, ring p, {oo})) p

o9 = 0

o9 : F

Over finite fields.

i10 : q = baseChange_13 p; zeros q

o11 = {2}

o11 : List

i12 : F = splittingField q; describe F

o13 = GF 4826809

i14 : #zeros_F q

o14 = 6

Ways to use zeros :

zeros(Ring,RingElement)
zeros(RingElement)

For the programmer

The object zeros is a method function with options.