In the previous chapter P2(x,n) was found to be a polynomial of degree 3n and divisible by a cubic polynomial G(x). In this chapter the division is defined in the finite field of the discriminant of G(x). Limits are placed on the degree 3n when the polynomial is to be completely factored in the field.
The decomposition of an N dimensional space into invariant sub-spaces is demonstrated using Groebner basis. Similar matrices are derived from the characteristic polynomials P2(x,n) and represented by symmetric geometric shapes.
Factoring P2(x, n) over a Finite Field
A computerized algebra algorithm is used to find polynomials and the integer coefficients a, b and n for which the polynomial F(x) = a x2n + bxn + Dn – (Bx2+Cx+D)n is divisible by G(x) = x3 +Bx2+Cx+D. Consequently, x is any of the real or complex roots of G such that G(x) = F(x) = 0. Also, the coefficients a, b are integers derived from the integer sequence associated with the cubic polynomial G. Higher order polynomials divisible by G(x) are also derived using a Groebner basis. It is shown that new recurrence relationships can be generated from the Groebner basis.
Chapter 21— Deducing Polynomial Division Algorithms