Polynomial

2020 Mathematics Subject Classification: Primary: 12E05 [MSN][ZBL]

An expression of the form

$$f(x,y,\dots,w)=$$

$$=Ax^ky^l\dotsm w^m+Bx^ny^p\dotsm w^q+\dots+Dx^ry^s\dotsm w^t,$$

where $x,y,\dots,w$ are variables and $A,B,\dots,D$ (the coefficients of the polynomial) and $k,l,\dots,t$ (the exponents of the powers, which are non-negative integers) are constants. The individual expressions

$$Ax^ky^l\dotsm w^m$$

are called the terms of the polynomial. The order of the terms, and also the order of the factors in each term, can be changed arbitrarily; in precisely the same way it is possible to introduce or omit terms with zero coefficients and, in each individual term, zero powers. When the polynomial has one, two or three terms it is called a monomial, binomial or trinomial.

With regard to the coefficients of a polynomial one assumes that they belong to a field, for example, the field of rational, real or complex numbers.

Two terms of a polynomial are called similar if the powers of the same variables in them are equal. Terms similar to each other,

$$A'x^ky^l\dotsm w^m,B'x^ky^l\dotsm w^m,\dots,D'x^ky^l\dotsm w^m,$$

can be replaced by one term

$$(A'+B'+\dots+D')x^ky^l\dotsm w^m$$

(reduction of similar terms). Two polynomials are called equal if, after reduction, all terms with non-zero coefficients are pairwise identical (but, possibly, written in a different order), and also if all the coefficients of both of these polynomials turn out to be zero. In the latter case the polynomial is called identically zero and is denoted by the symbol 0.

The sum of the powers of any term of a polynomial is called the degree of that term. If the polynomial is not identically zero, then among the terms with non-zero coefficients (it is assumed that similar terms have been reduced) there is at least one of highest degree: this highest degree is called the degree of the polynomial. The zero polynomial does not have a degree. A polynomial of degree zero reduces to a single term $A$ (a constant, not equal to zero).

A polynomial in the variables $x_1,\dots,x_n$ is called a symmetric polynomial if it is not changed by any permutation of the variables. A polynomial of which all terms have the same degree is called a homogeneous polynomial or a form; forms of the first, second or third degree are called linear, quadratic or cubic, and, according to the number of variables (two or three), they are called dyadic (binary) or triadic (ternary) (for example, $f(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2-x_1x_2-x_2x_3-x_1x_3$ is a ternary quadratic form).

The degree of a polynomial $f(x_1,\dots,x_n)$ with respect to one of its variables $x_i$, $i=1,\dots,n$, is the highest power with which $x_i$ occurs in a term of this polynomial (this degree may be zero). Of two terms of a polynomial the higher one (relative to a given numbering of the variables) is that for which the power of $x_1$ is higher, and if these powers are equal, that for which the power of $x_2$ is higher, etc. If all terms of a polynomial are ordered so that each term is lower than the preceding, then the terms are said to be lexicographically ordered. The term which then stands in the first place is called the highest term (or leading term). A polynomial of one variable with lexicographically ordered terms has the form

$$f(x)=a_0x^n+a_1x^{n-1}+\dots+a_n,$$

where $a_0,\dots,a_n$ are the coefficients.

The roots of a polynomial in one variable over a field $k$ are the solutions of the algebraic equation

$$f(x)=0.$$

The roots of a polynomial are related to its coefficients by Viète's formula (see Viète theorem).

The set of all possible polynomials in $n$ variables with coefficients from a given field forms a ring with respect to the naturally defined operations of addition and multiplication. The ring of polynomials in an infinite set of variables can also be considered. A ring of polynomials is an associative-commutative ring without zero divisors (that is, a product of non-zero polynomials cannot be 0).

If for two given polynomials $P$ and $Q$ there exists a polynomial $R$ such that $P=QR$, then one says that $P$ is divisible by $Q$; $Q$ is called the divisor and $R$ the quotient. If $P$ is not divisible by $Q$, but both polynomials contain the same variable, for example $x$, and the degree of $P$ relative to $x$ is $n$ and the degree of $Q$ relative to $x$ is $m$, $n\geq m\geq1$, then there are polynomials $p$, $R$ and $S$ such that $pP=QR+S$, where $p$ does not contain $x$ at all and $x$ occurs in $S$ with degree less than $m$. When $x$ is the only variable, then $p$ can be taken to be 1; in this case the operation of finding $R$ and $S$ from $P$ and $Q$ is called division with remainder; division with remainder can be carried out using the Horner scheme.

By repeated application of this operation it is possible to find the greatest common divisor of $P$ and $Q$, that is, the divisor of $P$ and $Q$ which is divisible by any common divisor of these polynomials (see Euclidean algorithm). Two polynomials with greatest common divisor equal to 1 are called coprime.

A polynomial which can be represented as a product of polynomials of smaller degree with coefficients from a given field is called reducible (over that field); otherwise it is called irreducible. The irreducible polynomials play a role in the ring of polynomials similar to that played by the prime numbers in the ring of integers. For example, the following theorem holds: If a product $PQ$ is divisible by an irreducible polynomial $R$ and $P$ is not divisible by $R$, then $Q$ must be divisible by $R$. Each polynomial of degree greater than zero splits over a given field into a product of irreducible factors in a unique way (up to factors of degree zero). For example, the polynomial $x^4+1$ is irreducible over the field of rational numbers, splits into two factors over the field of real numbers and into four factors over the field of complex numbers. In general, each polynomial of one variable $x$ with real coefficients splits over the field of real numbers into factors of the first and second degree, and over the field of complex numbers into factors of the first degree (cf. Algebra, fundamental theorem of). For two or more variables this is no longer true. Over any field $k$, for $n\geq2$ there are polynomials in $n$ variables that are irreducible over any extension of $k$. Such polynomials are called absolutely irreducible. For example, the polynomial $x^3+yz^2+z^3$ is irreducible over any number field.

If the variables $x,y,\dots,w$ are given numerical values (for example, real or complex), then the polynomial assumes a certain numerical value. Thus, a polynomial can be considered as a function of the corresponding variables. This function is continuous and differentiable for any values of the variables; it can be characterized as an entire rational function, that is, a function obtained from variables and constants (the coefficients) by performing in a specific order the operations of addition, subtraction and multiplication. Entire rational functions belong to the broader class of rational functions, where division is added to the list of operations: Any rational function can be represented as a quotient of two polynomials. Finally, rational functions are contained in the class of algebraic functions (cf. Algebraic function).

One of the most important properties of polynomials is that any continuous function on a compact subset of the complex plane can be approximated by a polynomial within arbitrarily small error (see Weierstrass theorem).

Special systems of polynomials, orthogonal polynomials, are used in approximation theory as a means of representing functions by series.

References

[1]	A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian)
[2]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
[3]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Springer (1988) pp. Chapt. 4–7 (Translated from French)

Comments

Polynomials over arbitrary rings $R$ are defined similarly, with the additional requirement that the coefficients (taken from $R$) are to commute with the variables $x,y,\dots,w$.

References

[a1]	S. Lang, "Algebra" , Addison-Wesley (1984)