# Specification of Minimal Polynomial function

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 evaluation precision field.

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 Minpoly precision field.

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
In general the output is  a multiple of the minimal polynomial of the algebraic number (i.e. not necessarily the minimal polynomial itself).
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