Input

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

Description

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

Output

The output is printed in
the ** Results **field and it consist
of

- 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 minimal polynomial

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.

On the 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.

Note

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.

Agnes Szanto

Last modified: Thu May 4 15:54:09 PDT 2000