AlgebraicNumberPolynomial¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

AlgebraicNumberPolynomial[a, x]

gives the polynomial in x corresponding to the AlgebraicNumber object a.

Notes

For a = AlgebraicNumber\[theta, {c0, c1, ..., cn}\], the result is the polynomial c0 + c1 x + ... + cn x^n, from which a is recovered by replacing x with theta. An integer or rational a is the constant polynomial and is returned unchanged; any other argument stays unevaluated. Threads over lists. Attributes: Listable, Protected.

Examples (7)¶

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (3)¶

In[1]:= AlgebraicNumberPolynomial[AlgebraicNumber[Sqrt[2] + Sqrt[3], {1, 2, 3, 4}], x]
Out[1]= 1 + 2 x + 3 x^2 + 4 x^3

Threads

In[2]:= AlgebraicNumberPolynomial[{2, AlgebraicNumber[Sqrt[2], {1, 2}]}, x]
Out[2]= {2, 1 + 2 x}

In[3]:= a = AlgebraicNumber[Sqrt[2 + Sqrt[3]], {1, 2, 3, 4}]; RootReduce[(AlgebraicNumberPolynomial[a, x] /. x -> Sqrt[2 + Sqrt[3]]) == a]
Out[3]= True

Worked examples (2)¶

In[4]:= AlgebraicNumberPolynomial[2, x]
Out[4]= 2

In[5]:= AlgebraicNumberPolynomial[1/2, x]
Out[5]= 1/2

Applications (2)¶

C0 + c1 x from {1, 2}

In[6]:= AlgebraicNumberPolynomial[AlgebraicNumber[Sqrt[2], {1, 2}], x]
Out[6]= 1 + 2 x

A rational is the constant polynomial

In[7]:= AlgebraicNumberPolynomial[7, x]
Out[7]= 7

Algorithm¶

algebraicnumberpolynomial.c — AlgebraicNumberPolynomial[a, x].

See algebraicnumberpolynomial.h for the contract. The coefficient vector of an AlgebraicNumber object is stored in the object itself, so producing the defining polynomial c0 + c1 x + ... + cn x^n is a purely structural read + build: no field arithmetic and no FLINT. The returned Plus/Times/Power tree is canonicalised by the evaluator on its next fixed-point step (zero terms drop, Times[1, x] folds to x, monomials sort by degree).

Implementation notes¶

Algorithm. AlgebraicNumberPolynomial[a, x] is a purely structural read and rebuild — no field arithmetic and no FLINT. The coefficient vector of an AlgebraicNumber object already lives in the object, so the defining polynomial is assembled directly:

An Integer, BigInt, or Rational a is the constant polynomial and is returned unchanged (expr_copy).
For a = AlgebraicNumber[theta, {c0, c1, …, cn}] whose coefficients are all exact rationals (Integer/BigInt/Rational — the forms canonicalisation stores), it builds c0 + c1 x + c2 x^2 + … + cn x^n as a Plus of Times/ Power terms. The generator theta is irrelevant here: a is recovered by substituting x -> theta. An empty coefficient list yields 0.
Any other argument routes an AlgebraicNumberPolynomial::naobj message through mth_message and stays unevaluated.

The returned Plus/Times/Power tree is left unevaluated; the evaluator canonicalises it on its next fixed-point step (zero terms drop, Times[1, x] folds to x, monomials sort by degree).

Data structures. Expr trees only. It reads the coefficient List out of the AlgebraicNumber[theta, {coeffs}] representation and emits a polynomial Expr; the anp_is_rational_coeff guard enforces the exact-rational invariant of a canonical object.

Complexity / limits. O(n) in the degree — one monomial per coefficient, with a single allocation sweep. Listable, Protected. It declines on anything that is not a rational or a well-formed AlgebraicNumber object (including an AlgebraicNumber with a symbolic or inexact coefficient).

Attributes: Listable, Protected.

References¶

See also: AlgebraicNumber, RootReduce

H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138 (Springer, 1993), §4.2 (power-basis representation of number-field elements).
Source: src/poly/algebraicnumberpolynomial.c
Specification: docs/spec/builtins/algebra.md
Tests: tests/test_algebraicnumberpolynomial.c

Notes & additional examples¶

Notes¶

AlgebraicNumberPolynomial[a, x] gives the polynomial in x whose coefficients are the stored coefficient list of the AlgebraicNumber object a: for a = AlgebraicNumber[theta, {c0, c1, …, cn}] the result is c0 + c1 x + … + cn x^n, and a is recovered by replacing x with theta. The generator theta plays no part — this is a purely structural read of the coefficient vector, with no field arithmetic.

An integer or rational a is the constant polynomial and is returned unchanged; any other argument stays unevaluated. Listable and Protected.