Subsections

• Creation of Structures
• Creation of Elements

There are two different ways to create polynomial rings, corresponding to the different internal representations (vector versus distributive --- see the introductory section): PolynomialRing(S) and PolynomialRing(S, n). The latter should be used to create multivariate polynomials; the former is recommended for univariate polynomials.

PolynomialAlgebra(P) : Rng -> RngUPol

PolynomialRing(P) : Rng -> RngUPol

Create a univariate polynomial ring over the ring S. The ring is regarded as an S-algebra via the usual identification of elements of S and the constant polynomials. The polynomials are stored in vector form, which allows fast arithmetic. It is not recommended to use this function recursively to build multivariate polynomial rings. The angle bracket notation can be used to assign names to the indeterminates: P<x> := PolynomialRing(R).

The easiest way to create polynomials in a given ring is to use the angle bracket construction to attach names to the indeterminates, and to use these names to express polynomials (see the examples). Below we list other options.

One(P) : RngUPol -> RngUPolElt

Identity(P) : RngUPol -> RngUPolElt

Zero(P) : RngUPol -> RngUPolElt

Representative(P) : RngUPol -> RngUPolElt

Return the indeterminate for the polynomial ring P, as an element of P.

Given a polynomial ring P=R[x] and elements a_0, ..., a_d coercible into the coefficient ring R, return the polynomial a_0 + a_1x_n + ... + a_dx_n^d as an element of P.

P ! s : RngUPol, [ RngElt ] -> RngPolElt

elt< P | s > : RngUPol, [ RngElt ] -> RngUPolElt

Coerce the element s into the polynomial ring P = R[x]. The following possibilities for s exist.

• s is an element of P: it is returned unchanged;
• s is an element of a ring that can be coerced into the coefficient ring R of P: the constant polynomial s is returned;
• s=sum_j s_jy^j is an element of a univariate polynomial ring whose coefficient ring elements s_j can be coerced into R: the polynomial sum_j r_jx^j is returned, where r_j is the result of coercing s_j into R;
• s is a sequence: if s is empty then the zero element of P is returned, and if it is non-empty but the elements of the sequence can be coerced into R then the polynomial sum_j s[j]x_n^(j - 1) is returned.

Note that constant polynomials cannot be coerced into their coefficient rings, but the function Coefficient can be used for that.

> P<x> := PolynomialRing(Integers());
> f := x^3+3*x+1;
> f;
x^3 + 3*x + 1

> P<x> := PolynomialAlgebra(Integers());
> f := elt< P | 2, 3, 0, 1 >;
> f;
x^3 + 3*x + 2
> P ! [ 2, 3, 0, 1 ];
x^3 + 3*x + 2

> Q := RationalField();
> Q ! [1, 2];
1/2
> P<x> := PolynomialRing(Q);
> P ! [1,2];
2*x + 1
> P ! Q ! [1,2];
1/2
> P ! [ [1,2], [2,3] ];
2/3*x + 1/2