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Math and science::Algebra::Aluffi

Polynomial rings

When polynomials are viewed as a type of notational container, they can be seen to have a ring structure.

Polynomial

A polynomial is a combination of "powers" of an indeterminate \( x \) with coefficients in a ring \( R \). It is written in the form:

\[ a_n x^n + ... + a_1 x + a_0 \]

or

\[ \sum_{i>0} a_i n^{i} \]

We require that there is an \( n \) above which all coefficients are zero.

Two polynomials are considered equal if all coefficients are equal.

The meaning of the operation \( \cdot \) in \( a_i \cdot x^i \) is not specified in the definition. Furthermore, the meaning of the \( + \) operation between elements is also not specified. The definition of equality cements the notational behaviour.

Notation

The set of polynomials with indeterminate \( x \) over ring \( R \) is denoted as \( R[x] \).

Polynomials as rings

A polynomial \( R[x] \) forms a ring (operations on the reverse side). As polynomials form rings, the construction can be nested such that polynomials in indeterminate \( y \) over \( R[x] \) for a ring. This is denoted as \( R[x][y] \) and \( R[x,y] \).

Polynomial rings can be generalized and be considered instances of monoid rings.


Addition and multiplication

Let \( f(x) \) and \( g(x) \) be two polynomials:

\[ f(x) = \sum_{i>0} a_i x^{i} \quad \text{and} \quad g(x) = \sum_{i>0} b_i x^{i} \]

Then we define addition, \( f(x) + g(x) \), as:

\[ f(x) + g(x) = \sum_{i>0} (a_i + b_i) x^{i} \]

and multiplication, \( f(x) \cdot g(x) \), as:

\[ f(x) \cdot g(x) = \sum_{i>0}\sum_{j+k=i} a_jb_k x^{j+k} \]

Note that multiplication has a convolution-like feel, which is discussed further below.

Ring of "Sequences"

Monoid rings feel like sequences of objects (or placeholders for objects). The objects are elements of a ring, which I'll call the inner-ring. And so, there is a natural way to add two sequences by simply delegating elementwise to the objects in the sequence. The placeholder analogy is nice as it encourages the visual of the placeholders being sometimes empty (containing the identity element of the inner-ring); and in fact, for polynomials, we will assert that only finite many placeholders are allowed to be non-empty.

As well as sequence addition, there is sequence multiplication. Sequence multiplication also delegates to the multiplication operation of the inner-ring; however, it does in a sort of convolution. What is somewhat magic is that this way of defining sequence multiplication allows the sequences themselves to form a ring.

Non-integer indexing

The sense of a sequence of placeholders, which evokes the sense of indexing with natural numbers, can be relaxed so that the "indexing" is done by a monoid: a set with an associative operation and an identity element. The convolution-like behaviour of the placeholder "sequence" multiplication is then less of a drag-and-flip behaviour and is instead powered by working backwards through the preimage of the monoid operation. Jumping back to indexing with natural numbers while holding on to the monoid viewpoint, we would see that the preimage of say 5 through the operation of \( + \) would give the set \( \{ (0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0) \} \).

Polynomials as infinite dimensional vectors

If the monoid used for "indexing" has infinite elements, such as if it is \( \mathbb{N} \), then there are infinite placeholders. In the case of polynomials, a restriction is in place that asserts that at most finite number of placeholders may be non-zero (non-identity) elements. In this way, polynomials can be considered a sort of infinite dimensional vector where only finite components may be non-zero.

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Don't forget the outer addition

The above discussion treats polynomials as simply notation. But in a polynomial all elements of the sequence are then added together—an infinite sum. A notation only perspective doesn't survive this sum, as the whole sequence can reduce to a single real.

Source

Aluffi p124-126