How to work with polynomial rings with real or complex coefficients

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The problem with real or complex fields is that their elements are approximate, and operations there are subject to rounding errors, which may accumulate with the number of performed operations. This is unacceptable situation for the ideals machinery, where accumulating errors destroy basic properties and make it impossible to perform basis operations (e.g., membership testing).

In other words, for ideals to work they have to be defined over exact fields/rings, which do not allow rounding errors of any kind. Unfortunately, real and complex fields cannot be implemented exactly. Nevertheless, there are suitable substitutes for real and complex fields:

• rational field QQ and rational complex field QQ[I];
• field of algebraic numbers QQbar and its real subfield AA.

We can approximate real and complex numbers with elements of AA and QQbar using algdep() function, where we can choose accuracy in the form of polynomial degree.

Clearly, rational fields QQ and QQ[I] are subfields of AA and QQbar, respectively, and appear when we approximate with polynomials of degree 1.

For example, we can approximate $\pi$ with a polynomial root of degree 5 as follows:

eps = 1e-6
pi_aa = AA.polynomial_root(algdep(pi,5),RIF(pi-eps,pi+eps))
print(pi_aa)
print(pi_aa.parent())

which gives

3.141592653589794?
Algebraic Real Field