Specification of Minimal Polynomial function
A real or complex number
specified in the alpha field.
It may be input as floating point approximation or as exact number.
In the latter case, it will first be evaluated
in floating point at precision specified in the evaluation
The Algebraic Test function
uses the linear
integer dependence algorithm to find a polynomial of degree specified
in the degree field (or less)
with integer coefficients such that the given input number approximates
one of its roots. The accuracy of the approximation is specified in the
Minpoly precision field.
The output is printed in
the Results field and it consist
In general the output is
a multiple of the minimal polynomial of the algebraic number (i.e. not
necessarily the minimal polynomial itself).
The minimal polynomial
of the input (or a multiple of it) together with the precision
The residual, i.e. the value
of the minimal polynomial at the input
The coefficient vector of the
It is because the linear
integer dependence algorithm finds a solution of minimal norm, and a polynomials with small norm can have factors with large norms. The minimal polynomial can
be obtained by factoring the result and choosing the right factor.
other hand returning a multiple of the minimal polynomial with small norm
sometimes helps: for example returning x^n-1 may be more informative than
the corresponding cyclotomic factor. For example, try exp(Pi/7*I).
The minimal polynomial has degree 6, but if we specify the degree to be
7, the algorithm returns a reducible polynomial.
If the input does not approximate
an algebraic number to the given precision then then the Minimal Polynomial
algorithm outputs a polynomial with coefficients of size roughly equal
to the given precision divided by the given degree. See also the
note on precision.
Back to IntegerRelations.
modified: Thu May 4 15:54:09 PDT 2000