In mathematics, a polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x² − x/4 + 7 is a polynomial, but x² − 4/x + 7x^3/2 is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not an integer (3/2). The term "polynomial" can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in polynomial time, which is used in computational complexity theory.

Polynomial comes from the Greek poly, "many" and medieval Latin binomium, "binomial".^[1][2][3] The word was introduced in Latin by Franciscus Vieta.^[4]

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

Overview[link]

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) that may be multiplied by a finite number of variables (usually represented by letters), also called indeterminates.^[5] Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x¹, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a (nonzero) constant term is 0. The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo of some prime number p. In most of the examples in this section, the coefficients are integers.

For example:

Failed to parse (Missing texvc executable; please see math/README to configure.): -5x^2y\,

is a term. The coefficient is –5, the variables are x and y, the degree of x is in the term two, while the degree of y is one.

The degree of the entire term is the sum of the degrees of each variable in it, so in this example the degree is 2 + 1 = 3.

Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:

Failed to parse (Missing texvc executable; please see math/README to configure.): \underbrace{_\,3x^2}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}} \underbrace{-_\,5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}} \underbrace{+_\,4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}.

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.

The commutative law of addition can be used to freely permute terms into any preferred order. In polynomials with one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, variable x, and exponent 2. In the second term, the coefficient is –5. The third term is a constant. Since the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.

Two terms with the same variables raised to the same powers are called "like terms", and they can be combined (after having been made adjacent) using the distributive law into a single term, whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0, in which case their combination just cancels out the terms. Polynomials can be added using the associative law of addition (which simply groups all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if

Failed to parse (Missing texvc executable; please see math/README to configure.): P=3x^2-2x+5xy-2 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): Q=-3x^2+3x+4y^2+8 \, ,

then

Failed to parse (Missing texvc executable; please see math/README to configure.): P+Q=3x^2-2x+5xy-2-3x^2+3x+4y^2+8 \,,

which can be simplified to

Failed to parse (Missing texvc executable; please see math/README to configure.): P+Q=x+5xy+4y^2+6 \,.

To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if

Failed to parse (Missing texvc executable; please see math/README to configure.): {\color{BrickRed}P {{=}} 2x + 3y + 5}

Failed to parse (Missing texvc executable; please see math/README to configure.): {\color{RoyalBlue}Q {{=}} 2x + 5y + xy + 1},

then

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{array}{rccrcrcrcr} {\color{BrickRed}P}{\color{RoyalBlue}Q}&{{=}}&&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}2x}) &+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}2x}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}1}) \\&&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}3y}\cdot {\color{RoyalBlue}xy})&+& ({\color{BrickRed}3y}\cdot{\color{RoyalBlue}1}) \\&&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}5y})&+& ({\color{BrickRed}5}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}1}) \end{array}

which can be simplified to

Failed to parse (Missing texvc executable; please see math/README to configure.): PQ=4x^2+21xy+2x^2y+12x+15y^2+3xy^2+28y+5 \,.

The sum or product of two polynomials is always a polynomial.

Alternative forms[link]

In general any expression can be considered a polynomial if it is built from variables and constants using only addition, subtraction, multiplication, and raising expressions to constant positive whole number powers. Such an expression can always be rewritten as a sum of terms. For example, (x + 1)³ is a polynomial; its standard form is x³ + 3x² + 3x + 1.

Division of one polynomial by another does not, in general, produce a polynomial, but rather produces a quotient and a remainder.^[6] A formal quotient of polynomials, that is, an algebraic fraction where the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. For example,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x^3}{12}

is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{1}{12}x^3

and Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{1}{12}
is just a constant. When this expression is used as a term, its coefficient is therefore Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{1}{12}

. For similar reasons, if complex coefficients are allowed, one may have a single term like Failed to parse (Missing texvc executable; please see math/README to configure.): (2+3i)x^3

even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term.

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over x^2 + 1} \,

is not a polynomial because it includes division by a non-constant polynomial.

Failed to parse (Missing texvc executable; please see math/README to configure.): ( 5 + y ) ^ x ,\,

is not a polynomial, because it contains a variable used as exponent.

Since subtraction can be replaced by addition of the opposite quantity, and since positive whole number exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and variables using only addition and multiplication.

Polynomial functions[link]

A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \,

for all arguments x, where n is a non-negative integer and a₀, a₁,a₂, ..., a_n are constant coefficients.

For example, the function ƒ, taking real numbers to real numbers, defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x^3 - x\,

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.\,

An example is also the function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=\cos(2\arccos(x))

which, although it doesn't look like a polynomial, is a polynomial function since for every x it is true that Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=2x^2-1
(see Chebyshev polynomials).

Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.

Polynomial equations[link]

A polynomial equation, also called algebraic equation, is an equation in which a polynomial is set equal to another polynomial.

Failed to parse (Missing texvc executable; please see math/README to configure.): 3x^2 + 4x -5 = 0 \,

is a polynomial equation. In case of a univariate polynomial equation, the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like (x + y)(x – y) = x² – y², where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.

Elementary properties of polynomials[link]

• A sum of polynomials is a polynomial.
• A product of polynomials is a polynomial.
• A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.
• The derivative of the polynomial a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ is the polynomial na_nx^n-1 + (n-1)a_n-1x^n-2 + ... + 2a₂x + a₁. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then ka_k should be interpreted as the sum of a_k with itself, k times. For example, over the integers modulo p, the derivative of the polynomial x^p+1 is the polynomial 0.
• If the division by integers is allowed in the set of coefficients, a primitive or antiderivative of the polynomial a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ is a_nxⁿ⁺¹/(n+1) + a_n-1xⁿ/n + ... + a₂x³/3 + a₁x²/2 + a₀x +c, where c is an arbitrary constant. Thus x²+1 is a polynomial with integer coefficients whose primitives are not polynomials over the integers. If this polynomial is viewed as a polynomial over the integers modulo 3 it has no primitive at all.

Polynomials serve to approximate other functions, such as sine, cosine, and exponential.

All polynomials have an expanded form, in which the distributive and associative laws have been used to remove all brackets and commutative law has been used to make the like terms adjacent and combine them. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. In the case of the field of complex numbers, the irreducible polynomials are linear. For example, the factored form of

Failed to parse (Missing texvc executable; please see math/README to configure.): 5x^3-5 \,

is

Failed to parse (Missing texvc executable; please see math/README to configure.): 5(x - 1)(x^2+x + 1),\,

over the integers and

Failed to parse (Missing texvc executable; please see math/README to configure.): 5(x - 1)(x+\frac{1+i\sqrt{3}}{2})(x+\frac{1-i\sqrt{3}}{2})\,

over the complex numbers.

Every polynomial in one variable is equivalent to a polynomial with the form

Failed to parse (Missing texvc executable; please see math/README to configure.): a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0.

This form is sometimes taken as the definition of a polynomial in one variable.

Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Actual evaluation is usually more efficient using the Horner scheme:

Failed to parse (Missing texvc executable; please see math/README to configure.): ((\cdots((a_n x + a_{n-1})x + a_{n-2})x + \cdots + a_3)x + a_2)x + a_1)x + a_0.\,

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and equals the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.

A system of polynomial equations is a set of equations in which each variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, in particular in linear algebra, methods are given for solving a system of linear equations in several unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. Overdetermined systems are common in practical applications. For example, one U.S. mapping survey used computers to solve 2.5 million equations in 400,000 unknowns.^[7]

Viète's formulas relate the coefficients of a polynomial to symmetric polynomial functions of its roots.

History[link]

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.

Notation[link]

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.^[8]

Solving polynomial equations[link]

Every polynomial P in x corresponds to a function, ƒ(x) = P (where the occurrences of x in P are interpreted as the argument of ƒ), called the polynomial function of P; the equation in x setting f(x) = 0 is the polynomial equation corresponding to P. The solutions of this equation are called the roots of the polynomial; they are the zeroes of the function ƒ (corresponding to the points where the graph of ƒ meets the x-axis). A number a is a root of P if and only if the polynomial x − a (of degree one in x) divides P. It may happen that x − a divides P more than once: if (x − a)² divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (x − a)^m divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P.

Some polynomials, such as x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

There is a difference between approximating roots and finding exact expressions for roots. Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times (see quadratic equation), and for polynomials of degree 3 or 4 similar formulas (using cube roots in addition to square roots) were found in the 16th century (see cubic function and quartic function for the formulas and Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general (finite) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). In 1830, Évariste Galois, studying the permutations of the roots of a polynomial, extended Abel-Ruffini theorem by showing that, given a polynomial equation, one may decide if it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and Group theory, two important branches of modern mathematics. Galois himself noted that the computations implied by his method were impracticable. Nevertheless formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Jenkins-Traub method, Laguerre's method, Durand–Kerner method or by some other root-finding algorithm.

For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero. However for certain sets of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions. If the number of solutions is finite, there are algorithms to compute the solutions. The methods underlying these algorithms are described in the article systems of polynomial equations. The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.

It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterisations of the roots of arbitrary polynomials are generalisations of the methods previously discovered to solve the quintic equation.

Properties of the roots[link]

The statistical properties of the roots of a random polynomial have been the subject of several studies. Let

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0

be a random polynomial. If the coefficients a_i are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be located on or close to the unit circle.

If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula^[9][10]

Failed to parse (Missing texvc executable; please see math/README to configure.): p( x ) = \frac { \sqrt{ A( x ) C( x ) - B( x )^2 }} {\pi A( x )}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): A( x ) = \sigma \sum { x^{ 2i } } = \sigma \frac{ x^{ 2n } - 1 } { x - 1 }

Failed to parse (Missing texvc executable; please see math/README to configure.): B( x ) = \frac{ 1 } { 2 } \frac{ d } { dt } A( x )

Failed to parse (Missing texvc executable; please see math/README to configure.): C( x ) = \frac{ 1 } { 4 } \frac{ d^2 } { dt^2 } A( x ) + \frac{ 1 } { 4x } \frac{ d } { dt } A( x )

When the coefficients are Gaussian distributed with a non zero mean and variance of σ, a similar but more complex formula is known.

Asymptotic results[link]

For large n, a number of asymptotic formulae are known. For a fixed x

Failed to parse (Missing texvc executable; please see math/README to configure.): p( x ) = \frac{ 1 } { \pi | 1 - x^2 | }

and

Failed to parse (Missing texvc executable; please see math/README to configure.): p( \pm 1 ) = \frac{ 1 } { \pi } \sqrt { \frac{ n^2 - 1 } { 12 } }

where p( x ) is the mean density of real roots. The expected number of real roots is

Failed to parse (Missing texvc executable; please see math/README to configure.): N_n = \frac{ 2 } { \pi } log( n ) + C + O( n^{ -2 } )

where C is a constant approximately equal to 0.6257358072 and O() is the order operator.

This result has bee shown by Kac, Erdos and others to be insensitive to the actual distribution of coefficients. Numerical testing of this formula has confirmed these earlier results.

Graphs[link]

A polynomial function in one real variable can be represented by a graph.

• The graph of the zero polynomial

f(x) = 0

is the x-axis.

• The graph of a degree 0 polynomial

f(x) = a₀, where a₀ ≠ 0,

is a horizontal line with y-intercept a₀

• The graph of a degree 1 polynomial (or linear function)

f(x) = a₀ + a₁x , where a₁ ≠ 0,

is an oblique line with y-intercept a₀ and slope a₁.

• The graph of a degree 2 polynomial

f(x) = a₀ + a₁x + a₂x², where a₂ ≠ 0

is a parabola.

• The graph of a degree 3 polynomial

f(x) = a₀ + a₁x + a₂x², + a₃x³, where a₃ ≠ 0

is a cubic curve.

• The graph of any polynomial with degree 2 or greater

f(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ , where a_n ≠ 0 and n ≥ 2

is a continuous non-linear curve.

The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

The illustrations below show graphs of polynomials.

• 

  Polynomial of degree 2:
  f(x) = x² - x - 2 = (x+1)(x-2)

• 

  Polynomial of degree 3:
  f(x) = x³/4 + 3x²/4 - 3x/2 - 2 = 1/4 (x+4)(x+1)(x-2)

• 

  Polynomial of degree 4:
  f(x) = 1/14 (x+4)(x+1)(x-1)(x-3) + 0.5

• 

  Polynomial of degree 5:
  f(x) = 1/20 (x+4)(x+2)(x+1)(x-1)(x-3) + 2

• 

  Polynomial of degree 6:
  f(x) = 1/30 (x+3.5)(x+2)(x+1)(x-1)(x-3)(x-4) + 2

• Septic Graph.gif

  Polynomial of degree 7:
  f(x) = (x-3)(x-2)(x-1)(x)(x+1)(x+2)(x+3)

Polynomials and calculus[link]

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomial functions. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Polynomial functions are also frequently used to interpolate functions.

Calculating derivatives and integrals of polynomial functions is particularly simple. For the polynomial function

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i=0}^n a_i x^i

the derivative with respect to x is

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i=1}^n a_i i x^{i-1}

and the indefinite integral is

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i=0}^n {a_i\over i+1} x^{i+1}+c.

Abstract algebra[link]

In abstract algebra, one distinguishes between polynomials and polynomial functions. A polynomial f in one variable X over a ring R is defined as a formal expression of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): f = a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X^1 + a_0X^0

where n is a natural number, the coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): a_0,\ldots,a_n

are elements of R, and X is a formal symbol, whose powers Xi are just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence Failed to parse (Missing texvc executable; please see math/README to configure.): (a_0, a_1, \ldots)

, where there is an n such that a_i = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a_iXⁱ is interpreted as a polynomial that has zero coefficients at all powers of X other than Xⁱ. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

Failed to parse (Missing texvc executable; please see math/README to configure.): a X^k \; b X^l = ab X^{k+l}

   for all elements a, b of the ring R and all natural numbers k and l.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. The map from R to R[X] sending r to rX⁰ is an injective homomorphism of rings, by which R is viewed as a subring of R[X]. If R is commutative, then R[X] is an algebra over R.

One can think of the ring R[X] as arising from R by adding one new element X to R, and extending in a minimal way to a ring in which X satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is Xr = rX). To do this, one must add all powers of X and their linear combinations as well.

Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring R[X] over the real numbers by factoring out the ideal of multiples of the polynomial X² + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

If R is commutative, then one can associate to every polynomial P in R[X], a polynomial function f with domain and range equal to R (more generally one can take domain and range to be the same unital associative algebra over R). One obtains the value f(r) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for X. And it should be noted that if R is not commutative, there is no (well behaved) notion of polynomial function at all.

Divisibility[link]

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[X] such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotient can be computed using the Horner scheme.

If F is a field and f and g are polynomials in F[X] with g ≠ 0, then there exist unique polynomials q and r in F[X] with

Failed to parse (Missing texvc executable; please see math/README to configure.): f = q \, g + r

and such that the degree of r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[X] is a Euclidean domain.

Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as polynomials which cannot be factorized into the product of two non constant polynomials. Any polynomial may be decomposed into the product of a constant by a product of irreducible polynomials. This decomposition is unique up to the order of the factors and the multiplication of any constant factors by a constant (and division of the constant factor by the same constant. When the coefficients belong to a finite field or are rational numbers, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials. These algorithms are not practicable for hand written computation, but are available in any Computer algebra system (see Berlekamp's algorithm for the case in which the coefficients belong to a finite field or the Berlekamp–Zassenhaus algorithm when working over the rational numbers ^[11]). Eisenstein's criterion can also be used in some cases to determine irreducibility.

See also: Greatest common divisor of two polynomials.

Classifications[link]

Polynomials are classified according to many different properties.

Number of variables[link]

One classification of polynomials is based on the number of distinct variables. A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result, for instance, from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any variables at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of variables allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is common, also, to say simply "polynomials in x, y, and z", listing the variables allowed. In this case, xy is allowed.

Degree[link]

A second major way of classifying polynomials is by their degree. Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term.

Usually, a polynomial of degree n, for n greater than 3, is called a polynomial of degree n, although the phrases quartic polynomial and quintic polynomial are sometimes used. The use of names for degrees greater than 5 is even less common. The names for the degrees may be applied to the polynomial or to its terms. For example, in Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 + 2x + 1

the term Failed to parse (Missing texvc executable; please see math/README to configure.): 2x
is a first degree term in a second degree polynomial.

In the context of polynomial interpolation there is some ambiguity when combining the two classifications above. For example, a bilinear interpolant, being the product of two univariate linear polynomials, is bivariate but is not linear; similar ambiguity affects the bicubic interpolant.

The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either –1 or –∞).^[12] These conventions are important when defining Euclidean division of polynomials. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots.

If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in x of degree n then takes the general form

Failed to parse (Missing texvc executable; please see math/README to configure.): c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0

where

c_n ≠ 0, c_n-1, ..., c₂, c₁ and c₀ are constants, the coefficients of this polynomial.

Here the term c_nxⁿ is called the leading term and its coefficient c_n the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.

Note that apart from the leading coefficient c_n (which must be non-zero or else the polynomial would not be of degree n) this general form allows coefficients to be zero; when this happens the corresponding term is zero and may be removed from the sum without changing the polynomial. It is nevertheless common to refer to c_i as the coefficient of xⁱ, even when c_i happens to be 0, so that xⁱ does not really occur in any term; for instance one can speak of the constant term of the polynomial, meaning c₀ even if it is zero.

In the case of polynomials in more than one variable, a polynomial is called homogeneous of degree n if all its terms have degree n. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): x^3y^2 + 7x^2y^3 - 3x^5

is homogeneous.

Coefficients[link]

Another classification of polynomials is by the kind of constant values allowed as coefficients. One can work with polynomials with integer, rational, real, or complex coefficients, and in abstract algebra polynomials with many other types of coefficients can be defined, such as integers modulo p. As in the classification by number of variables, when working with coefficients for a given set, such as the complex numbers, coefficients from any subset are allowed. Thus Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 + 3x -5

is a polynomial with integer coefficients, but it is also a polynomial with complex coefficients, because the integers are a subset of the complex numbers.

Number of non-zero terms[link]

Polynomials may also be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and so on. (Some authors use "monomial" to mean "monic monomial".^[13])

Polynomials associated to other objects[link]

Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.

Extensions of the concept of a polynomial[link]

Polynomials can involve more than one variable, in which they are called multivariate. Rings of polynomials in a finite number of variables are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. These rings can alternatively be constructed by repeating the construction of univariate polynomials with as coefficient ring another ring of polynomials: thus the ring R[X,Y] of polynomials in X and Y can be viewed as the ring (R[X])[Y] of polynomials in Y with as coefficients polynomials in X, or as the ring (R[Y])[X] of polynomials in X with as coefficients polynomials in Y. These identifications are compatible with arithmetic operations (they are isomorphisms of rings), but some notions such as degree or whether a polynomial is considered monic can change between these points of view. One can construct rings of polynomials in infinitely many variables, but since polynomials are (finite) expressions, any individual polynomial can only contain finitely many variables.

A binary polynomial where the second variable takes the form of an exponential function applied to the first variable, for example P(X,e^X ), may be called an exponential polynomial.

Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.

Quotients of polynomials are called rational expressions (or rational fractions), and functions that evaluate rational expressions are called rational functions. Rational fractions are formal quotients of polynomials (they are formed from polynomials just as rational numbers are formed from integers, writing a fraction of two of them; fractions related by the canceling of common factors are identified with each other). The rational function defined by a rational fraction is the quotient of the polynomial functions defined by the numerator and the denominator of the rational fraction. The rational fractions contain the Laurent polynomials, but do not limit denominators to powers of a variable. While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not null. A rational function produces rational output for any rational input for which it is defined; this is not true of other functions such as trigonometric functions, logarithms and exponential functions.

Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like real numbers cannot), but the rules for manipulating their terms are the same as for polynomials.

See also[link]

• Binomial
• Lill's method
• List of polynomial topics
• Polynomials on vector spaces
• Power series

Notes[link]

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2010)

References[link]

• R. Birkeland. Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen. Mathematische Zeitschrift vol. 26, (1927) pp. 565–578. Shows that the roots of any polynomial may be written in terms of multivariate hypergeometric functions.
• F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions.
• F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1892 edition.
• K. Mayr. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. Monatshefte für Mathematik und Physik vol. 45, (1937) pp. 280–313.
• H. Umemura. Solution of algebraic equations in terms of theta constants. In D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, 1984.

External links[link]

Wikimedia Commons has media related to: Polynomials

Look up polynomial in Wiktionary, the free dictionary.

• List of Calculators for Quadratic through Sextic equations
• Euler's work on Imaginary Roots of Polynomials at Convergence
• Characteristics of polynomials
• Free online polynomial root finder for both real and complex coefficients

Aphex Twin
In Turin, Italy in March 2007.
Background information
Birth name	Richard David James
Born	(1971-08-18) 18 August 1971 (age 40) Limerick, Ireland
Origin	Lanner, Cornwall, United Kingdom
Genres	Electronic, ambient, IDM, acid techno
Occupations	Musician, composer, remixer
Instruments	Synthesizers, piano, electronics, harpsichord, organ, percussion, laptop, softsynth, turntables, drum machine, guitar, bass guitar, computer, sequencer, sampler
Years active	1985–present
Labels	Warp Records Sire Records Rephlex Records
Associated acts	Universal Indicator, Mike & Rich

Richard David James (born 18 August 1971), best known under the pseudonym Aphex Twin, is a British electronic musician and composer. He founded the record label Rephlex Records in 1991 with Grant Wilson-Claridge. He was described by The Guardian as "the most inventive and influential figure in contemporary electronic music."^[1]

Aphex Twin has also recorded music under the aliases AFX, Blue Calx, Bradley Strider, Caustic Window, DJ Smojphace, GAK, Martin Tressider, Polygon Window, Power-Pill, Prichard D. Jams, Q-Chastic, Tahnaiya Russell, The Dice Man, Soit-P.P., and speculatively The Tuss.

Aphex Twin has released recordings on Rephlex, Warp, R&S, Sire, Mighty Force, Rabbit City, and Men Records.

History[link]

Early years[link]

Richard David James was born to Welsh parents Lorna and Derek James in St. Munchin's Limerick Regional Maternity Hospital, Ireland. He grew up in Lanner, Cornwall, enjoying, along with two older sisters, a "very happy" childhood during which they, according to James, "were pretty much left to do what [they] wanted."^[2] He "liked growing up there, being cut off from the city and the rest of the world".^[3] James attended Redruth School, located in Redruth, Cornwall.^[4]

According to musician Benjamin Middleton, James started producing music at the age of 12.^[5] As a teenager he was a disc jockey at the Shire Horse Inn in St Ives, with Tom Middleton at the Bowgie Inn in Crantock, and also along the numerous beaches around Cornwall. James studied for a National Diploma in Engineering from 1988 to 1990 (aged 16 to 18) at Cornwall College. Talking about his studies, James has said that "music and electronics went hand in hand."^[6] James graduated from college; according to an engineering lecturer, however, he often had his headphones on during practical lessons, "no doubt thinking through the mixes he'd be working on later".^[7]

Early career: early 1990s[link]

In 1989 James met and befriended Grant Wilson-Claridge when working as a DJ on alternate weeks at the Bowgie pub, near Newquay in Cornwall. Wilson-Claridge was intrigued by James's sets and was surprised to discover that James was playing tapes of his own music.

James' first release was the 12-inch EP Analogue Bubblebath on Mighty Force Records in 1991. It was first released under the moniker Aphex Twin, later changed to AFX. The track "En Trance to Exit" was made in collaboration with Tom Middleton, also known as Schizophrenia.^[8] The EP got on the playlist of Kiss FM, an influential London radio station, which helped the release to become a success.^[9]

In 1991, James and Wilson-Claridge founded Rephlex Records to promote "innovation in the dynamics of Acid — a much-loved and -misunderstood genre of house music forgotten by some and indeed new to others, especially in Britain."^[10]

Between 1991 and 1993, James released two Analogue Bubblebath EPs as AFX, and an EP under Bradley Strider, Bradley's Beat. James moved to London to take an electronics course at Kingston Polytechnic, but at the time admitted to David Toop that his "electronics studies were already slipping away as a career in the techno business took precedence." After quitting his course, James remained in London and released a number of albums and EPs on Warp Records and other labels under many aliases including AFX, Polygon Window, and Power-Pill. A number of James' tracks (released under the aliases Blue Calx, The Dice Man, and others) appeared on various compilations. Local legend has it that James lived on the roundabout in Elephant and Castle, South London during his early years in the capital - in fact he lived in a nearby disused bank.^[2][11]

Gaining success: 1992–1995[link]

	"Ageispolis" noicon Sample of "Ageispolis" from the album Selected Ambient Works 85-92.
Problems listening to this file? See media help.

	"Ventolin (Video Version)" noicon Sample of "Ventolin (Video Version)" from the album ...I Care Because You Do.
Problems listening to this file? See media help.

The first full-length Aphex Twin album, Selected Ambient Works 85–92, was released in 1992 on R&S Records. It received high ratings and praise from critics. John Bush of Allmusic described it as a "watershed of ambient music".^[12] In 2002, Rolling Stone wrote of the album: "Aphex Twin expanded way beyond the ambient music of Brian Eno by fusing lush soundscapes with oceanic beats and bass lines."^[13] Pitchfork Media's review called it, "among the most interesting music ever created with a keyboard and a computer."^[14] Critics also noted that the songs were recorded on cassette and that the sound quality was "relatively poor".

In 1992, he also released the EPs Xylem Tube EP and Digeridoo (first played by DJ Colin Faver on London's Kiss FM) as Aphex Twin, as Power-Pill the Pac-Man EP based on the arcade game Pac-Man, and two of his four Caustic Window EPs. "Digeridoo" reached #55 in the UK Singles Chart, and was later described as foreshadowing drum and bass by Rolling Stone.^[15] "Digeridoo" was recorded initially for the benefit of FIZZ-BOMB (at the Shire Horse, St Ives, Cornwall). These early releases came out on Rephlex Records, Mighty Force of Exeter, and R&S Records of Belgium.^[16]

In 1993, James released Analogue Bubblebath 3. He also released a single titled "On"; his second Bradley Strider EP, Bradley's Robot; two more Caustic Window EPs; and his first releases on Warp Records, Surfing on Sine Waves and "Quoth" under the alias Polygon Window.

Warp Records pressed and released a follow-up to Selected Ambient Works 85-92, Selected Ambient Works Volume II in 1994. The sound was much less beat and melody-driven than the previous volume. All of the track names were described with pie chart symbols, each of which was meant to be paired with a corresponding image in the album jacket, with exception for one song, which was named "Blue Calx". To decipher song titles, listeners had to compare the length of each track with the size of each pie chart symbol. For example, the first title, which is often labelled cliffs, is realised by pairing the first symbol with the first image, which is that of a rocky cliffside.^[17] James claimed in The Wire magazine and other media that these songs were inspired by lucid dreams and synaesthesia. Other releases are a fourth Analogue Bubblebath; GAK, derived from early demos sent to Warp Records; and Classics, a compilation album that includes "Digeridoo" and the Xylem Tube EP.

For his 1995 release, ...I Care Because You Do, James used an image of his face for the album cover, a motif that would continue on many of his later records. The album was a compilation of songs composed between 1990 and 1994 and represented a mish-mash of Aphex Twin's various music styles. This was James's last record of the 1990s to use mostly analogue synthesizers. He commissioned Western classical-music composer Philip Glass to create an orchestral version of one of the songs from this album, "Icct Hedral", which appeared on the Donkey Rhubarb EP.^[18]

Jungle, DSP, and laptops: 1995–1999[link]

In 1995 (primarily with Hangable Auto Bulb, a near anagram of Analogue Bubblebath), James began releasing more material composed on computers, combining a jungle sound with nostalgic childhood themes and computer-generated acid lines.

James's early adoption of software synthesizers predated the later popularity of using computers to make music. In the mid-to-late 1990s, his music gradually gained more mainstream appeal, as he released Richard D. James Album and Expert Knob Twiddlers (a collaboration with fellow dance producer µ-ziq) in 1996, "Come to Daddy" in 1997 (#36 on UK charts) and "Windowlicker" in 1999 (#16 on UK charts). Two pop songs that heavily use Digital Signal Processing (DSP) techniques, "Come to Daddy" and "Windowlicker" were shown on MTV and became cover features for music magazines such as NME. The videos for both singles were directed by British artist Chris Cunningham and caused controversy on release because of their disturbing images and themes.

In 1998, Aphex Twin's track "[rhubarb]" (from Selected Ambient Works Volume II) appeared as part of the BBC's digital widescreen test transmission, which was on loop for many months as the service was introduced across the UK.^[19]

Prepared piano, laptops, and more DSP: 2000–2003[link]

In 2001 Aphex Twin released drukqs, a two-CD album that featured computer controlled piano songs influenced by Erik Satie and John Cage. Many of the tracks' names are written in the Cornish language (i.e., 'jynweythek' translatable as 'machinemusic'). Also included were abrasive, fast, and meticulously programmed computer-made songs. Rolling Stone described the piano songs as "aimlessly pretty".^[20] Some reviewers concluded that drukqs was released as a contract breaker with Warp Records—a credible guess, as James's next big release was released on his own Rephlex label. James claimed to interviewers he had accidentally left an MP3 player labelled "Aphex Twin—unreleased tracks" on a plane, containing a large set of new songs, and rushed the album release to preempt an Internet leak.^[21] He also released a short EP titled 2 Remixes By AFX the same year as Drukqs. It featured two remixes, the originals being from 808 State and DJ Pierre. In addition, there is one untitled third track that consists of an SSTV image, which consists of mostly of high-pitched sounds, which can be decoded to a viewable image by proper software such as MultiMode (for Macintosh) or MMSSTV (for Windows).

Synthesizers and drum machines: 2004–present[link]

In late 2004, rumours of James's return to an acid techno-based sound were realised with the Analord series, an 11-part series of EPs with 42 total tracks, averaging 2–4 tracks per EP. The series was created by playing and sequencing analogue and digital electronic music equipment such as synthesizers and drum machines. The series was recorded on magnetic tape and then later pressed on vinyl.

James was meticulous about the whole process of recording, mastering, and pressing. James has said Rephlex Records was strict on quality control, trying out various pressing-plant companies until they felt it sounded perfect. To James's ears, vinyl or tape is better than digital, as no two copies are the same. However, label co-owner Grant Wilson-Claridge convinced James to release a CD compilation, Chosen Lords, which included 10 selected tracks from the Analord series. For the Analord series James used his collection of vintage synthesizer and drum machines, some of which were quite rare by that time. Some of the record inserts have pictures of rare synthesizers like the Synton Fenix, and the notoriously difficult-to-program Roland MC-4 sequencer, as well as the legendary Roland TB-303.

In 2007 media reports indicated that Aphex Twin was recording under a new alias called The Tuss. Rephlex Records has denied that Aphex Twin is The Tuss, but Aphex Twin fans and the media have ignored Rephlex's denial and The Tuss is treated as yet another Aphex Twin project;^[22][23] further evidence being provided by the fact that all Tuss tracks are published in the BMI Repertoire under the name JAMES RICHARD DAVID.^[24]

Some of Aphex Twin's music was featured in the award winning documentary about the atrocities of shark finning, Sharkwater.

In March 2009 Steve Beckett of Warp Records announced a new album for a possible release but the album never materialised.^[25] However, 20 more tracks were added to the Analord series in December 2009, only available through download from the Rephlex Records website.

In an October 2010 interview with British magazine Another Man, James stated that he had completed 6 albums, one of which was a remake of the unreleased Melodies from Mars, originally produced around the time of Richard D. James Album.^[26] In June 2011, he spoke to Spanish magazine EL PAIS. When asked about the 6 albums, he answered 'More than 10 or 11 are already compiled, and many more songs are orphans.' He also revealed that a new album 'will show in a while' and that the reason there has been so long since his last album was that he was waiting for a divorce from his wife.^[27]

Background details[link]

	"Jynweythek" noicon Sample of "Jynweythek" from the album Drukqs.
Problems listening to this file? See media help.

	"Vordhosbn" noicon Sample of "Vordhosbn" from the album Drukqs.
Problems listening to this file? See media help.

	"Fenix Funk 5" noicon Sample of "Fenix Funk 5" from the EP Analord 10.
Problems listening to this file? See media help.

	"VBS.Redlof. B" noicon Sample of "VBS.Redlof. B" from the EP Analord 11.
Problems listening to this file? See media help.

The Aphex Twin name[link]

The name Aphex is derived from Aphex Systems Limited, a brand of audio signal processing equipment (itself derived from "analogue effects").^{[dubious – discuss]} It is used with permission, as acknowledged on the back sleeves of Richard D. James Album and drukqs. He has explained in interviews that the Twin portion of his moniker is used in memory of his brother, also named Richard James, who died at birth in 1968.

Artwork[link]

James's own face, grinning or distorted in some way, is a common theme in his album covers, his music videos, and the songs themselves. The cover of I Care Because You Do features a painting of James, while the cover of Richard D. James Album presents a close-up photograph. His face is transplanted onto other peoples' bodies in the music videos of "Come to Daddy" and "Windowlicker". Near the end of the second track of the "Windowlicker" single (commonly referred to as "Equation"), a photo of James' face is embedded as a steganogram which is revealed when run through spectral analysis.^[28] Another embedded image of James, along with collaborator Tom Jenkinson, appears in the third track of 2 Remixes by AFX, "Bonus High Frequency Sounds", encoded in SSTV format, with text relating to the release.

James has used his own photography for some of his releases, such as the elaborate album sleeve for Selected Ambient Works Volume II.

Braindance[link]

Richard's own Rephlex Records label, which he co-owns with Grant Wilson-Claridge, coined the term Braindance in 1991 to describe Aphex Twin's music.^[22][29][30] Rephlex Records' official definition of Aphex Twin and his followers' music is quoted as follows: "Braindance is the genre that encompasses the best elements of all genres, e.g traditional, classical, electronic music, popular, modern, industrial, ambient, hip-hop, electro, house, techno, breakbeat, hardcore, ragga, garage, drum and bass, etc."^[31]

Braindance applies to forward-thinking electronic music that can appeal to the mind as well as the desire to dance and party. Examples including Ed-DMX's Breakin' records label, µ-ziq's Planet-mu label, the Aphex Twin EP Come to Daddy and Astrobotnia Parts 1, 2 & 3.^[32]

Influences[link]

At age 17, Richard D. James mentioned these influences: "Phonic Boy, Computer World, Mental Telepathy, Industrial Inc., Tomita, Tangerine Dream". Mixmaster Morris mentions on the "I Luv AFX" BBC Radio 1 Breezeblock session that James' preferred moniker while working as a DJ in Cornwall was Phonic Boy on Dope. More recently, he has said that he gets inspiration from "everyday sounds that can be emulated/reconstructed electronically, quality techno, especially from Europe, which overshadows the current hardcore pop crap." When asked about what is next for electronic music, he said "acid-techno, ambient-techno." Avant-garde music has been an influence on James, including artists such as Kraftwerk, Can, Tangerine Dream, Tod Dockstader, Xenakis, Piero Umiliani, Bernard Parmegiani, Karlheinz Stockhausen,^[33] John Cage, and the French composer Erik Satie for his piano works and his ideas for furniture music.

In acknowledgment of another influence, James released Music from the BBC Radiophonic Workshop, a compilation of music recorded by the pioneers of BBC Radiophonic Workshop, which included artists such as Delia Derbyshire,^[34] on his own label, Rephlex Records.

Aphex Twin has said, "I don't really like rock & roll." Despite this, he has mentioned being a fan of Led Zeppelin, citing their use of breakbeats as an influence, as well as Pink Floyd for their psychedelic music.^[35]

Many songs include sounds from and references to the ZX Spectrum. For instance, "Carn Marth" from Richard D. James Album includes the tape-loading noise of the game Sabre Wulf.

Influence on others[link]

The London Sinfonietta has performed arrangements of Aphex Twin tracks.^[36] In 2005, the orchestra Alarm Will Sound released Acoustica: Alarm Will Sound Performs Aphex Twin. The album consists of acoustic arrangements of some of James's electronic tracks. He has also had an influence on rock bands like Radiohead.^[37] However, he has dismissed the notion of going on tour with them: "I wouldn't play with them since I don't like them."^[35] Despite his previous comments, he premiered new music alongside Radiohead guitarist/composer Jonny Greenwood in a 2011 collaboration with Polish composer Krzysztof Penderecki.^[38] The animator David Firth has much of his work influenced and soundtracked by Aphex Twin.^[39]

Intelligent Dance Music[link]

The term 'intelligent dance music', or IDM, was coined in August 1993 by the IDM mailing list based at hyperreal.org, as a term to describe the sound pioneered by the Warp Records Artificial Intelligence Series. The series featured Aphex Twin tracks (under a different pseudonym), as well as early productions from labelmates such as Autechre and LFO. The usage of this term spread in the USA and on internet messageboards, but the term is still a source of controversy and derision amongst the artists and fans, including Aphex Twin. As of October 2011 the list is still active.

I just think it's really funny to have terms like that. It's basically saying, 'this is intelligent and everything else is stupid.' It's really nasty to everyone else's music. (laughs) It makes me laugh, things like that. I don't use names. I just say that I like something or I don't.

—Aphex Twin^[40]

Aphex Twin refers to his own breakbeat tracks as breakbeat, jungle, or drum and bass interchangeably. Other genres that he indulges in include acid, ambient, ambient techno, noise, and many more.

Equipment[link]

During an interview with Future Music, Richard said he likes using Ableton Live but prefers Liveslice for beat editing/stretching. Asked what equipment did use, he responded "Raveolution 309, the Raven Max, MC-909 limited edition, Quasimidi Van Helden, MAM Freebass 383, Roland DJ-70, E-15, SP-808, Akai S3200, Behringer MX602A and all the Behringer effects that copy other things."^[44]

Aphex Twin's press[link]

James described himself in The Guardian as follows: "I'm just some irritating, lying, ginger kid from Cornwall who should have been locked up in some youth detention centre. I just managed to escape and blag it into music."^[1]

James said he composed ambient techno music at the age of 13; he claims to have "over 100 hours" of unreleased music, and to have made his own software to compose with, including algorithmic processes that automatically generate beats and melodies; in addition he claims to experience synaesthesia, and that he is able to incorporate lucid dreaming into the process of making music.^[45]

He lives in southeast London in a converted bank, which was formerly the Bank of Cyprus and then HSBC. Contrary to popular opinion, however, he does not own the silver structure in the centre of the roundabout at Elephant and Castle. This is, in fact, the Michael Faraday Memorial, containing a power transformer for the Northern Line, which James jokingly claimed to be buying in an interview with The Face magazine in 2001.^[2] Some of these rumours are hard to confirm as he has been known to spread mistruths in the prankster tradition, making such claims as only sleeping two to three hours a night.^[46]

Stockhausen vs. The Technocrats[link]

In November 1995, The Wire published an article titled "Advice to Clever Children". In the process of producing the interview, a package of tapes containing music from several artists, including Aphex Twin, was sent to Karlheinz Stockhausen.

Stockhausen commented:

I heard the piece Aphex Twin of Richard James (sic) carefully: I think it would be very helpful if he listens to my work "Song of the Youth," which is electronic music, and a young boy's voice singing with himself. Because he would then immediately stop with all these post-African repetitions, and he would look for changing tempi and changing rhythms, and he would not allow to repeat any rhythm if it varied to some extent and if it did not have a direction in its sequence of variations.^[47]

Aphex Twin, a fan of Stockhausen, responded:

"I thought he should listen to a couple of tracks of mine: "Digeridoo", then he'd stop making abstract, random patterns you can't dance to".^[47]

ZX81 competition[link]

Richard claims to have produced sound on a Sinclair ZX81 (a machine with no sound hardware) at the age of 11:

When I was 11, I won 50 pounds in a competition for writing this program that made sound on a ZX81. You couldn't make sound on a ZX81, but I played around with machine code and found some codes that retuned the TV signal so that it made this really weird noise when you turned the volume up.^[2]

Discography[link]

Studio albums (as Aphex Twin)

• Selected Ambient Works 85–92 (1992)
• Selected Ambient Works Volume II (1994)
• ...I Care Because You Do (1995)
• Richard D. James Album (1996)
• Drukqs (2001)

References[link]

External links[link]

Wikimedia Commons has media related to: Aphex Twin

• Aphex Twin at Warp Records
• Aphex Twin at the Open Directory Project
• Aphex Twin discography at Discogs.