Polynomial/Advanced

In algebra, a polynomial is, roughly speaking, a formal expression obtained from constants (called coefficients) and one or more variables by making a finite number of additions, subtractions and multiplications. For instance, $x^{2}-2x+1$ is a polynomial of one variable with integer coefficients, whereas $x^{2}+{\sqrt {2}}xy+y^{2}$ is a polynomial of two variables with real number coefficients.

Two binary operations, called addition and multiplication are defined on the set of polynomials. These operations, in turn, are defined using addition and multiplication operations on the coefficients. Thus, the most general context where one can define polynomials that can be added and multiplied using the usual definitions is when the coefficients are drawn from a set with two binary operations. To ensure that addition and multiplication of polynomials have useful properties, typically the coefficients are restricted to be in a commutative ring with identity. This is by far the most useful type of polynomial, and will be the type of polynomial considered in this article. However, various noncommutative analogs of polynomials, including rings of twisted polynomials (in which constants do not necessarily commute with variables during polynomial multiplication) and non-commuting polynomials (where the variables do not commute with each other) are useful in some specialized contexts.

Definition

Polynomials in one variable

There are many possible equivalent approaches to defining polynomials. For instance, they can be defined as the convolution algebra of the monoid of non-negative powers of the generator X of a cyclic group. This method also allows one to define non-commuting polynomial rings, and to view polynomials in one variable as a special case. Alternatively, polynomials can be defined as infinite sequences of coefficients such that all but a finite number of coefficients are equal to zero. This approach is useful because it allows one to view a polynomial ring as a subring of a ring of formal power series. This is the approach that will be used in this article.

Let us consider some expressions like $X^{2}-2X+1$, ½X³+X−√2 , or $2X^{5}-3X^{2}+1$. We can write all of them as follows:

$X^{2}-2X+1=1+(-2)X+1X^{2}+0X^{3}+0X^{4}+\cdots ,$

${\frac {1}{2}}X^{3}+X-{\sqrt {2}}=-{\sqrt {2}}+1X+0X^{2}+{\frac {1}{2}}X^{3}+0X^{4}+\cdots ,$

$2X^{5}-3X^{2}+1=1+0X+(-3)X^{2}+0X^{3}+0X^{4}+2X^{5}+0X^{6}+\cdots .$

This suggests that a polynomial can be entirely defined by giving a sequence of numbers, which are called its coefficients, all of them being zero from some rank. For instance the three polynomials above can be written respectively $(1,-2,1,0,0,\cdots )$, $\left(-{\sqrt {2}},1,0,{\frac {1}{2}},0,\cdots \right)$, and $(1,0,-3,0,0,2,0,\cdots )$, the dots meaning the sequence continues with an infinity of zeros. This leads to the definition below.

Definition. A polynomial $P$, over the ring $R$ is a sequence $P=\left(a_{0},a_{1},a_{2},\cdots ,a_{n},\cdots \right)$ of elements of $R$, called the coefficients of $P$, this sequence containing only a finite number of nonzero terms. The rank of the last nonzero term is called the degree of the polynomial.

Hence, the degrees of the three polynomials given above are respectively 2, 3 and 5. By convention, the degree of $(0,0,\cdots )$ is set to $-\infty$.

This definition may surprise the reader, because in reality, one thinks of a polynomial as an expression of the form $a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}$ rather than $\left(a_{0},a_{1},a_{2},\cdots ,a_{n},\cdots \right)$. We will progressively show how to return to this usual way of writing a polynomial. First, we identify any element $a_{0}$ of the ring to the polynomial $\left(a_{0},0,0,\cdots \right)$. For instance, we write only $7$ instead of the cumbersome $\left(7,0,0,\cdots \right)$, (or in the familiar fashion $7+0X+0X^{2}+\cdots$).

Secondly, we merely denote by $X$ the polynomial

$X=\left(0,1,0,0,\cdots \right)$.

This is natural, as in the familiar fashion this sequence corresponds to $0+1X+0X^{2}+0X^{3}+\cdots$ It remains to give a sense to $X^{2}$, $X^{3}$, etc. This will be made in the next two subsections.

Operations

We now define addition and multiplication of polynomials.

Addition

With the traditional notation, if we have $P=2X^{5}-3X^{2}+1$ and $Q=-X^{5}+4X^{4}+2X^{2}-1$, we want to have $P+Q=(2-1)X^{5}+4X^{4}+(-3+2)X^{2}+1-1=X^{5}+4X^{4}-X^{2}$, that is, one wants to add coefficients separately for each degree. This leads to the formal definition below.

Definition. Given two polynomials $P=\left(a_{0},a_{1},a_{2},\dots \right)$ and $Q=\left(b_{0},b_{1},b_{2},\dots \right)$, the sum $P+Q$ is defined by $P+Q=\left(a_{0}+b_{0},a_{1}+b_{1},a_{2}+b_{2},\dots \right)$.

Multiplication

Multiplication is harder to define. Let us begin with an example using traditional notation. For $P=X^{2}+X-2$ and $Q=2X^{2}-3X+1$, we want to have

$PQ=X^{2}\left(2X^{2}-3X+1\right)+X\left(2X^{2}-3X+1\right)-2\left(2X^{2}-3X+1\right)$; $PQ=2X^{4}+(-3+2)X^{3}+(1-3-2\cdot 2)X^{2}+(1-2\cdot (-3))X-2$; $PQ=2X^{4}-X^{3}-6X^{2}+7X-2\$.

One can observe that the coefficient of say, $X^{2}$, is obtained by adding $1\cdot 1$, $1\cdot (-3)$ and $-2\cdot 2$, that is, by adding all the $a_{i}b_{j}$ so that $i+j=2$, where the $a_{i}$ denote the coefficients of $P$ and the $b_{j}$ those of $Q$. Those mechanics lead to give the definition below.

Definition. Given two polynomials $P=\left(a_{0},a_{1},a_{2},\cdots \right)$ and $Q=\left(b_{0},b_{1},b_{2},\cdots \right)$, the product $PQ$ is defined by $PQ=\left(c_{0},c_{1},c_{2},\cdots \right)$, where for every index $k$, the coefficient $c_{k}$ is given by $c_{k}=\sum _{i+j=k}a_{i}b_{j}$.

The reader which is upset by those cumbersome notations should just retain that this definition allows to multiply polynomials (considered as mere sequences of coefficients) as one is used to do in elementary algebra (using the traditional notation, as in the example). The only striking fact is that in our construction, $X$ does not represent a number, but a pure abstract entity for which we have defined some rules of calculation.

The algebra $R[X]$

With the definition above, one can verify that the product of the polynomial $X=\left(0,1,0,0,\dots \right)$ by itself, that is $X^{2}$, is the sequence $X^{2}=\left(0,0,1,0,0,\dots \right)$. More generally, for each natural number $n$, one can verify that the $n$-th power of $X$ is given by $X^{n}=\left(0,\dots ,0,1,0,0,\dots \right)$, where the $1$ is the coefficient of index $n$ and all other coefficients are zeros. In particular, we have the usual convention $X^{0}=\left(1,0,0,\dots \right)$, which we identified to the constant $1$.

Now, any polynomial $P=\left(a_{0},a_{1},a_{2},\dots ,a_{n},0,0,\dots \right)$ is exactly equal to $a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}$, where the addition and the powers (which are mere repetitions of multiplications) are defined as in the preceding subsections. Our whole construction legitimates the traditional notation, and from now on, we will only use the later, with which calculations use natural rules of elementary algebra. It is however important to remember that the "variable" $X$ did not denote some number in our construction, but a particular sequence of coefficients. We have succeeded in defining polynomials in a purely formal manner.

Polynomials versus polynomial functions

It may be convenient to think of a polynomial as a function of its variables, that is, $x\mapsto x^{2}-2x+1$ or $(x,y)\mapsto x^{2}+y^{2}$. Such a function is called a polynomial function. But in reality, both concepts are different, the unspecified variables being purely formal entities when one thinks of an abstract polynomial, whereas they are meant to be replaced by any number when one thinks of a function. The distinction is important in abstract algebra, because what we have called "constant numbers" is more generally replaced by any ring, and for some rings the two concepts cannot be identified. There is not such a problem with polynomials over rings of usual numbers like integers, rational, real or complex numbers. Still it is important to understand that calculations with polynomials can be conceived in an only formal way, without giving any special ontological status to the variables. To make the distinction clear, it is common in algebra to denote the abstract variables with capital letters ($X$, $Y$, etc.), while variables of functions are still denoted with lowercase letters. We will use this convention in what follows.