# Polynomials

## Polynomials Topics

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### Horner's rule

A rule for polynomial computation which both reduces the number of necessary multiplications and results in less numerical instability due to potential subtraction of one large number from another. The rule simply factors out powers of , giving

### Gauss's polynomial theorem

Let be an integer polynomial. The can be factored into a product of two polynomials of lower degree with rational coefficients iff it can be factored into a product of integer polynomials of lower degree.

### Coefficient

A multiplicative factor (usually indexed) such as one of the constants in the polynomialIn this polynomial, the monomials are , , ..., , and 1, and the single variable is .

### Polynomial identity

Polynomial identities involving sums and differences of like powersinclude(1)(2)(3)(4)(5)(6)(7)(8)which are the polynomial versions of the so-called binomialnumbers.Further identities include(9)(10)The identity(11)was used by Lamé in his proof that Fermat's last theorem was true for .

### Reciprocal polynomial

Given a polynomial in a single complex variable with complex coefficientsthe reciprocal polynomial is defined bywhere denotes the complex conjugate.

### Christoffel formula

Let be orthogonal polynomials associated with the distribution on the interval . Also let(for ) be a polynomial of order which is nonnegative in this interval. Then the orthogonal polynomials associated with the distribution can be represented in terms of the polynomials asIn the case of a zero of multiplicity , we replace the corresponding rows by the derivatives of order 0, 1, 2, ..., of the polynomials , ..., at .

### Mason's theorem

Let there be three polynomials , , and with no common factors such thatThen the number of distinct roots of the three polynomials is one or more greater than their largest degree. The theorem was first proved by Stothers (1981).Mason's theorem may be viewed as a very special case of a Wronskian estimate (Chudnovsky and Chudnovsky 1984). The corresponding Wronskian identity in the proof by Lang (1993) isso if , , and are linearly dependent, then so are and . More powerful Wronskian estimates with applications toward Diophantine approximation of solutions of linear differential equations may be found in Chudnovsky and Chudnovsky (1984) and Osgood (1985).The rational function case of Fermat'slast theorem follows trivially from Mason's theorem (Lang 1993, p. 195).

### Rational zero theorem

If the coefficients of the polynomial(1)are specified to be integers, then rational roots must have a numerator which is a factor of and a denominator which is a factor of (with either sign possible). This follows since a polynomial of polynomial order with rational roots can be expressed as(2)where the roots are , , ..., and . Factoring out the s,(3)Now, multiplying through,(4)where we have not bothered with the other terms. Since the first and last coefficients are and , all the rational roots of equation (1) are of the form [factors of ]/[factors of ].

### Luk&aacute;cs theorem

Let be an th degree polynomial which is nonnegative in . Then can be represented in the form(1)where , , , and are real polynomials whose degrees do not exceed .

### Bring quintic form

A Tschirnhausen transformation can be used to take a general quintic equation to the formwhere may be complex.

### Laurent polynomial

A Laurent polynomial with coefficients in the field is an algebraic object that is typically expressed in the formwhere the are elements of , and only finitely many of the are nonzero. A Laurent polynomial is an algebraic object in the sense that it is treated as a polynomial except that the indeterminant "" can also have negative powers.Expressed more precisely, the collection of Laurent polynomials with coefficients in a field form a ring, denoted , with ring operations given by componentwise addition and multiplication according to the relationfor all and in the integers. Formally, this is equivalent to saying that is the group ring of the integers and the field . This corresponds to (the polynomial ring in one variable for ) being the group ring or monoid ring for the monoid of natural numbers and the field ...

### Bouniakowsky conjecture

Define a Bouniakowsky polynomial as an irreducible polynomial with integer coefficients, degree , and . The Bouniakowsky conjecture states that is prime for an infinite number of integers (Bouniakowsky 1857). As an example of the greatest common divisor caveat, the polynomial is irreducible, but always divisible by 2.Irreducible degree 1 polynomials () always generate an infinite number of primes by Dirichlet's theorem. The existence of a Bouniakowsky polynomial that can produce an infinitude of primes is undetermined. The weaker fifth Hardy-Littlewood conjecture asserts that is prime for an infinite number of integers .Various prime-generating polynomialsare known, but none of these always generates a prime (Legendre).Worse yet, it is unknown if a general Bouniakowsky polynomial will always produce at least 1 prime number. For example, produces no primes until , 764400, 933660, ... (OEIS A122131)...

### Rational function

A quotient of two polynomials and ,is called a rational function, or sometimes a rational polynomial function. More generally, if and are polynomials in multiple variables, their quotient is called a (multivariate) rational function. The term "rational polynomial" is sometimes used as a synonym for rational function. However, this usage is strongly discouraged since by analogy with complex polynomial and integer polynomial, rational polynomial should properly refer to a polynomial with rational coefficients.A rational function has no singularities other than poles in the extended complex plane. Conversely, if a single-values function has no singularities other than poles in the extended complex plane, then it is a rational function (Knopp 1996, p. 137). In addition, a rational function can be decomposed into partial fractions (Knopp 1996, p. 139)...

### Bivariate polynomial

A bivariate polynomial is a polynomial in two variables.Bivariate polynomials have the formA homogeneous bivariate polynomial, also called a binaryform, has the formwhere is the degree of . The binary form above is often represents as .

### Bieberbach conjecture

The th coefficient in the power series of a univalent function should be no greater than . In other words, ifis a conformal mapping of a unit disk on any domain and and , then . In more technical terms, "geometric extremality implies metric extremality." An alternate formulation is that for any schlicht function (Krantz 1999, p. 150).The conjecture had been proven for the first six terms (the cases , 3, and 4 were done by Bieberbach, Lowner, and Garabedian and Schiffer, respectively), was known to be false for only a finite number of indices (Hayman 1954), and true for a convex or symmetric domain (Le Lionnais 1983). The general case was proved by Louis de Branges (1985). de Branges proved the Milin conjecture, which established the Robertson conjecture, which in turn established the Bieberbach conjecture (Stewart 1996).authorresultBieberbach (1916)Löwner (1923)Garabedian and Schiffer (1955)Pederson (1968), Ozawa..

### Kronecker's algorithm

A polynomial factorization algorithm that proceeds by considering the vector of coefficients of a polynomial , calculating , constructing the Lagrange interpolating polynomials from the conditions and , and checking to see which are factorizations.

### Bernstein's polynomial theorem

If is a trigonometric polynomial of degree satisfying the condition where is arbitrary and real, then .

### Bernstein's inequality

Let be a polynomial of degree with derivative . Thenwhere

### Kac formula

The expected number of real zeros of a random polynomial of degree if the coefficients are independent and distributed normally is given by(1)(2)(Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by(3)(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand (left) and numerical values of (red curve in right plot) for small . The first few values are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955, ....As ,(4)where(5)(6)(OEIS A093601; top curve in right plot above).The initial term was derived by Kac (1943).

### Beauzamy and d&eacute;got's identity

For , , , and polynomials in variables(1)where(2) is the differential operator, is the Bombieri inner product, and(3)

### Primitive polynomial

A primitive polynomial is a polynomial that generates all elements of an extension field from a base field. Primitive polynomials are also irreducible polynomials. For any prime or prime power and any positive integer , there exists a primitive polynomial of degree over GF(). There are(1)primitive polynomials over GF(), where is the totient function.A polynomial of degree over the finite field GF(2) (i.e., with coefficients either 0 or 1) is primitive if it has polynomial order . For example, has order 3 since(2)(3)(4)Plugging in to equation (◇), the numbers of primitive polynomials over GF(2) are(5)giving 1, 1, 2, 2, 6, 6, 18, 16, 48, ... (OEIS A011260) for , 2, .... The following table lists the primitive polynomials (mod 2) of orders 1 through 5.primitive polynomials123, 4, 5, , , , , Amazingly, primitive polynomials over GF(2) define a recurrence relation which can be used to obtain a new pseudorandom bit from the preceding ones...

### Primitive part

The primitive part of a polynomial is , where is the content.For a general univariate polynomial , the Wolfram Language function FactorTermsList[poly, x] returns a list of three elements, the first being the integer content , the second being the polynomial content, i.e., a primitive (with respect to all variables) polynomial that does not depend on and which divides all coefficients of , and the third element being the primitive part of . The original polynomial is then the product of these three parts. For example, FactorTermsList[9E x^3 + 3E, x] returns 3, E, 1 + 3x^2.

### Prime divisor

If is a nonconstant integer polynomial and is an integer such that is divisible by the prime , that is called a prime divisor of the polynomial (Nagell 1951, p. 81). Every integer polynomial which is not a constant has an infinite number of prime divisors (Nagell 1951, p. 82).

Let be a field of finite characteristic . Then a polynomial is said to be additive iff for . For example, is additive for , sinceA more interesting class of additive polynomials known as absolutely additive polynomials are defined on an algebraic closure of . For example, for any such , is an absolutely additive polynomial, since , for , ..., . The polynomial is also absolutely additive.Let the ring of polynomials spanned by linear combinations of be denoted . If , then is not commutative.Not all additive polynomials are in . In particular, if is an infinite field, then a polynomial is additive iff . For be a finite field of characteristic , the set of absolutely additive polynomials over equals , so the qualification "absolutely" can be dropped and the term "additive" alone can be used to refer to an element of .If is a fixed power and , then is a ring of polynomials in . Moreover, if , then for all . In this case, is said to be a -linear polynomial.The..

### Power polynomial

The power polynomials are an associated Sheffer sequence with(1)giving generating function(2)and exponential generating function(3)and binomial identity(4)

### Ac method

The method is an algorithm for factoring quadratic polynomials of the form with integer coefficients. As its name suggests, the crux of the algorithm is to consider the multiplicative factors of the product of the coefficients and . More precisely, the goal is to find an integer pair and satisfying and simultaneously, whereby one can rewrite in the form(1)and to factor the remaining four-term polynomial by grouping into a product of linear factors with integer coefficients.For example, consider the polynomial having coefficients , , and . To begin the factorization, consider the product . By observation, while ; in particular, this guarantees that can be rewritten so that(2)This four-term expression for can be factored by grouping:(3)and so(4)One can easily see that the above method generalizes to certain polynomials of the form for positive integers , though the result will be a factorization into pairs of polynomials of degree which aren't..

### Invertible polynomial

A polynomial admitting a multiplicative inverse. In the polynomial ring , where is an integral domain, the invertible polynomials are precisely the constant polynomials such that is an invertible element of . In particular, if is a field, the invertible polynomials are all constant polynomials except the zero polynomial.If is not an integral domain, there may be in invertible polynomials that are not constant. In , for instance, we have:which shows that the polynomial is invertible, and inverse to itself.

### Polynomial sequence

A sequence of polynomials , for , 1, 2, ..., where is exactly of degree for all .

### Abel's lemma

The pure equationof prime degree is irreducible over a field when is a number of the field but not the th power of an element of the field.Jeffreys and Jeffreys (1988) use the term "Abel's lemma" for another lemma related to Abel's uniform convergence test.

### Polynomial roots

A root of a polynomial is a number such that . The fundamental theorem of algebra states that a polynomial of degree has roots, some of which may be degenerate. For example, the roots of the polynomial(1)are , 1, and 2. Finding roots of a polynomial is therefore equivalent to polynomial factorization into factors of degree 1.Any polynomial can be numerically factored, althoughdifferent algorithms have different strengths and weaknesses.The roots of a polynomial equation may be found exactly in the Wolfram Language using Roots[lhs==rhs, var], or numerically using NRoots[lhs==rhs, var]. In general, a given root of a polynomial is represented as Root[^n+a[n-1]^(n-1)+...+a[0]&, k], where , 2, ..., is an index identifying the particular root and the pure function polynomial is irreducible. Note that in the Wolfram Language, the ordering of roots is different in each of the commands Roots, NRoots, and Table[Root[p, k], k, n].In the Wolfram..

### Integer polynomial

A polynomial of the formhaving coefficients that are all integers. An integer polynomial gives integer values for all integer arguments of (Nagell 1951, p. 73). The set of integer polynomials is denoted . Integer polynomials are sometimes also called "integral polynomials," although this usage should be deprecated due to its confusing use of the term "integral" as an adjective.An integer polynomial is called primitive if the greatest common divisor .

### Abel's irreducibility theorem

If one root of the equation , which is irreducible over a field , is also a root of the equation in , then all the roots of the irreducible equation are roots of . Equivalently, can be divided by without a remainder,where is also a polynomial over .

### Eisenstein's irreducibility criterion

Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial is irreducible in the polynomial ring .The polynomialwhere for all and (which means that the degree of is ) is irreducible if some prime number divides all coefficients , ..., , but not the leading coefficient and, moreover, does not divide the constant term .This is only a sufficient, and by no means a necessary condition. For example, the polynomial is irreducible, but does not fulfil the above property, since no prime number divides 1. However, substituting for produces the polynomial , which does fulfill the Eisenstein criterion (with ) and shows the polynomial is irreducible.

### Sextic equation

The general sextic equationcan be solved in terms of Kampé de Fériet functions, and a restricted class of sextics can be solved in terms of generalized hypergeometric functions in one variable using Klein's approach to solving the quintic equation.

### Resolvent cubic

For a given monic quarticequation(1)the resolvent cubic is the monic cubic polynomial(2)where the coefficients are given in terms of the by(3)(4)(5)The roots , , and of are given in terms of the roots , , , and of by(6)(7)(8)The resolvent cubic of a quartic equation can be used to solve for the roots of the quartic in terms of the roots of the cubic, which can in turn be solved for using the cubic equation.

### Zero polynomial

The constant polynomial whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials.The degree of the zero polynomial is undefined, but many authors conventionally set it equal to or . In the Wolfram Language, Exponent[0, x] returns -Infinity.

### Icosahedral equation

There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron. Consider an icosahedron centered , oriented with -axis along a fivefold () rotational symmetry axis, and with one of the top five edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest icosahedral equation is defined by projecting the vertices of the icosahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the height above..

### Polynomial remainder

The remainder obtained when dividing a polynomial by another polynomial . The polynomial remainder is implemented in the Wolfram Language as PolynomialRemainder[p, q, x], and is related to the polynomial quotient byFor example, the polynomial remainder of and is , corresponding to polynomial quotient .

### Polynomial quotient

The quotient of two polynomials and , discarding any polynomial remainder. Polynomial quotients are implemented in the Wolfram Language as PolynomialQuotient[p, q, x], and are related to the polynomial remainder byFor example, the polynomial quotient of and is , leaving remainder .

### Vieta's formulas

Let be the sum of the products of distinct polynomial roots of the polynomial equation of degree (1)where the roots are taken at a time (i.e., is defined as the symmetric polynomial ) is defined for , ..., . For example, the first few values of are(2)(3)(4)and so on. Then Vieta's formulas states that(5)The theorem was proved by Viète (also known as Vieta, 1579) for positive roots only, and the general theorem was proved by Girard.This can be seen for a second-degree polynomialby multiplying out,(6)(7)so(8)(9)(10)(11)(12)(13)Similarly, for a third-degree polynomial,(14)(15)so(16)(17)(18)(19)(20)(21)(22)

### Polynomial order

The highest order power in a univariate polynomial is known as its order (or, more properly, its polynomial degree). For example, the polynomialis of order , denoted . The order of a polynomial is implemented in the Wolfram Language as Exponent[poly, x].It is preferable to use the word "degree" for the highest exponent in a polynomial, since a completely different meaning is given to the word "order" in polynomials taken modulo some integer (where this meaning is the one used in the multiplicative order of a modulus). In particular, the order of a polynomial with is the smallest integer for which divides (Lidl and Niederreiter 1994). For example, in the finite field GF(2), the order of is 31, sinceThis concept is closely related to that of the multiplicativeorder.If is an irreducible polynomial of degree , then its order has to divide the order of the multiplicative group in the corresponding field extension, i.e., for modulus..

### Polynomial norm

For a polynomial(1)several classes of norms are commonly defined. The -norm is defined as(2)for , giving the special cases(3)(4)(5)Here, is called the polynomial height. Note that some authors (especially in the area of Diophantine analysis) use as a shorthand for and as a shorthand for , while others (especially in the area of computational complexity) used to denote the -norm and (Zippel 1993, p. 174).Another class of norms is the -norms, defined by(6)for , giving the special cases(7)(8)(9)(Borwein and Erdélyi 1995, p. 6).

### Polynomial height

The -polynomial norm defined for a polynomial byNote that some authors (especially in the area of Diophantine analysis) use as a shorthand for , while others (especially in the area of computational complexity) used to denote the -norm (Zippel 1993, p. 174).

### Gr&ouml;bner basis

A Gröbner basis for a system of polynomials is an equivalence system that possesses useful properties, for example, that another polynomial is a combination of those in iff the remainder of with respect to is 0. (Here, the division algorithm requires an order of a certain type on the monomials.) Furthermore, the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gröbner basis is equivalent to Gaussian elimination.The algorithm for computing Gröbner bases is known as Buchberger's algorithm. Calculating a Gröbner basis is typically a very time-consuming process for large polynomial systems (Trott 2006, p. 37).Gröbner bases are pervasive in the construction of symbolic algebra algorithms, and Gröbner bases with respect to lexicographic order are very useful for solving equations and for elimination..

### Tetrahedral equation

The tetrahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider a tetrahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top three edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest tetrahedral equation is defined by projecting the vertices of the tetrahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not..

### Polynomial factorization

A factor of a polynomial of degree is a polynomial of degree less than which can be multiplied by another polynomial of degree less than to yield , i.e., a polynomial such thatFor example, sinceboth and are factors of .Polynomial factorization can be performed in the Wolfram Language using Factor[poly]. Factorization over an algebraic number field is implemented as Factor[poly, Extension -> ext].The coefficients of factor polynomials are often required to be real numbers or integers but could, in general, be complex numbers. The fundamental theorem of algebra states that a polynomial of degree has values (some of which are possibly degenerate) for which . Such values are called polynomial roots.The average number of factors of a polynomial of degree with integer coefficients in the range has been considered by Schinzel (1976), Pinner and Vaaler (1996), Bérczes and Hajdu (1998), and Dubickas (1999)...

### Term

In logic, a term is a variable, constant, or the result of acting on variables and constants by function symbols.In algebra, a term is a product of the form (in the univariate case), or more generally of the form (in the multivariate case) in a polynomial (Becker and Weispfenning 1993, p. 188).The word "term" is also used commonly to mean a summand of a polynomial including its coefficient (more properly called a monomial), or the corresponding quantity in a series (i.e., a series term).One term is said to divide another if the powers of its variables are no greater than the corresponding powers in the second monomial. For example, divides but does not divide . A term is said to reduce with respect to a polynomial if the leading term of that polynomial divides . For example, reduces with respect to because divides , and the result of this reduction is , or . A polynomial can therefore be reduced by reducing its terms beginning with the greatest..

### Synthetic division

Synthetic division is a shortcut method for dividing two polynomials which can be used in place of the standard long division algorithm. This method reduces the dividend and divisor polynomials into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.For an example of synthetic division, consider dividing by . First, if a power of is missing from either polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the respective polynomial. In this case the term is missing from the dividend while the term is missing from the divisor; therefore, is added between the quintic and the cubic terms of the dividend while is added between the cubic and the linear terms of the divisor:(1)and(2)respectively.Next, all the variables and their exponents () are removed from the dividend, leaving instead..

### Fundamental theorem of algebra

Every polynomial equation having complex coefficients and degree has at least one complex root. This theorem was first proven by Gauss. It is equivalent to the statement that a polynomial of degree has values (some of them possibly degenerate) for which . Such values are called polynomial roots. An example of a polynomial with a single root of multiplicity is , which has as a root of multiplicity 2.

### Symmetric polynomial

A symmetric polynomial on variables , ..., (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy(1)where and being an arbitrary permutation of the indices 1, 2, ..., .For fixed , the set of all symmetric polynomials in variables forms an algebra of dimension . The coefficients of a univariate polynomial of degree are algebraically independent symmetric polynomials in the roots of , and thus form a basis for the set of all such symmetric polynomials.There are four common homogeneous bases for the symmetric polynomials, each of which is indexed by a partition (Dumitriu et al. 2004). Letting be the length of , the elementary functions , complete homogeneous functions , and power-sum functions are defined for by(2)(3)(4)and for by(5)where is one of , or . In addition, the monomial functions are defined as(6)where is the set of permutations..

### Polynomial discriminant

A polynomial discriminant is the product of the squares of the differences of the polynomial roots . The discriminant of a polynomial is defined only up to constant factor, and several slightly different normalizations can be used. For a polynomial(1)of degree , the most common definition of the discriminant is(2)which gives a homogenous polynomial of degree in the coefficients of .The discriminant of a polynomial is given in terms of a resultant as(3)where is the derivative of and is the degree of . For fields of infinite characteristic, so the formula reduces to(4)The discriminant of a univariate polynomial is implemented in the Wolfram Language as Discriminant[p, x].The discriminant of the quadratic equation(5)is given by(6)The discriminant of the cubic equation(7)is given by(8)The discriminant of a quartic equation(9)is(10)(Schroeppel 1972)...

### Fundamental continuity theorem

Given two univariate polynomials of the same order whose first coefficients (but not the first ) are 0 where the coefficients of the second approach the corresponding coefficients of the first as limits, the second polynomial will have exactly roots that increase indefinitely. Furthermore, exactly roots of the second will approach each root of multiplicity of the first as a limit.

### Polynomial degree

The highest power in a univariate polynomial is known as its degree, or sometimes "order." For example, the polynomialis of degree , denoted . The (structural) degree of a polynomial is implemented in the Wolfram Language as Exponent[poly, x]. Richardson's theorem proves that it is recursively undecidable to determine the degree of an arbitrary polynomial.

### Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by(1)The individual summands with the coefficients (usually) included are called monomials (Becker and Weispfenning 1993, p. 191), whereas the products of the form in the multivariate case, i.e., with the coefficients omitted, are called terms (Becker and Weispfenning 1993, p. 188). However, the term "monomial" is sometimes also used to mean polynomial summands without their coefficients, and in some older works, the definitions of monomial and term are reversed. Care is therefore needed in attempting to distinguish these conflicting usages.The highest power in a univariate polynomial is calledits order, or sometimes its degree.Any polynomial with can be expressed as(2)where the product..

### Subresultant

Subresultants can be viewed as a generalization of resultants, which are the product of the pairwise differences of the roots of polynomials. Subresultants are the most commonly used tool to compute the resultant or greatest common divisor of two polynomials with coefficients in an integral ring. Subresultants for a few simple pairs of polynomials include(1)(2)(3)The principal subresultants of two polynomials can be computed using the Wolfram Language function Subresultants[poly1, poly2, var]. The first subresultants of two polynomials and , both with leading coefficient one, are zero when and have common roots.

### Orthogonal polynomials

Orthogonal polynomials are classes of polynomials defined over a range that obey an orthogonality relation(1)where is a weighting function and is the Kronecker delta. If , then the polynomials are not only orthogonal, but orthonormal.Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization.A table of common orthogonal polynomials is given below, where is the weighting function and(2)(Abramowitz and Stegun 1972, pp. 774-775).polynomialintervalChebyshev polynomial of the first kindChebyshev polynomial of the..

### Stable type

A polynomial equation whose roots all have negative real parts. For a real quadratic equationthe stability conditions are . For a real cubic equationthe stability conditions are and .

### Octahedral equation

The octahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider an octahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top four edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest octahedral equation is defined by projecting the vertices of the octahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the..

### Stable polynomial

A real polynomial is said to be stable if all its roots lie in the left half-plane. The term "stable" is used to describe such a polynomial because, in the theory of linear servomechanisms, a system exhibits unforced time-dependent motion of the form , where is the root of a certain real polynomial . A system is therefore mechanically stable iff is a stable polynomial.The polynomial is stable iff , and the irreducible polynomial is stable iff both and are greater than zero. The Routh-Hurwitz theorem can be used to determine if a polynomial is stable.Given two real polynomials and , if and are stable, then so is their product , and vice versa (Séroul 2000, p. 280). It therefore follows that the coefficients of stable real polynomials are either all positive or all negative (although this is not a sufficient condition, as shown with the counterexample ). Furthermore, the values of a stable polynomial are never zero for and have..

### Normal polynomial

In every residue class modulo , there is exactly one integer polynomial with coefficients and . This polynomial is called the normal polynomial modulo in the class (Nagell 1951, p. 94).

### Sparse polynomial square

A sparse polynomial square is a square of a polynomial that has fewer terms than the original polynomial . Examples include Rényi's polynomial(1)(Rényi 1947, Coppersmith and Davenport 1991), which has 29 terms and whose square has 28, Choudhry's polynomial(2)(Coppersmith and Davenport 1991), which has 18 terms and whose square has 17, and(3)(Coppersmith and Davenport 1991; Trott 2004, p. 276), which has 13 terms and whose square has 12.In fact, Coppersmith and Davenport (1991) found eight polynomials of degree 13 having sparse squares (of degree 12),(4)where six of the values are rational: , , , , , and (Abbott 2002). Using Gröbner bases, Abbott (2002) showed that no polynomial of degree less than 12 has a sparse square, but was not able to demonstrate that these examples are exhaustive...

### De moivre's quintic

The quintic equation(1)is sometimes known as de Moivre's quintic (Spearman and Williams 1994). It has solutions(2)for , 1, 2, 3, 4, where and are given by the simultaneous equations(3)(4)(Spearman and Williams 1994).

### Sch&ouml;nemann's theorem

If the integral coefficients , , ..., of the polynomialare divisible by a prime number , while the free term is not divisible by , then is irreducible in the natural rationality domain.

### Schinzel's hypothesis

If , ..., are irreducible polynomials with integer coefficients such that no integer divides , ..., for all integers , then there should exist infinitely many such that , ..., are simultaneously prime.If Schinzel's hypothesis is true, then it follows that all positive integers can be represented in the form with and prime. In addition, it would follow that there are an infinite number of numbers such that , where is the number of divisors of and is the sum of divisors, since the conjecture implies that there are infinitely many primes for which is prime, for such , and , so is in the sequence (D. Hickerson, pers. comm., Jan. 24, 2006).Conroy (2001) verified the conjecture to .

### Ruffini's rule

Ruffini's rule a shortcut method for dividing a polynomial by a linear factor of the form which can be used in place of the standard long division algorithm. This method reduces the polynomial and the linear factor into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.Note that Ruffini's rule is a special case of the more generalized notion of synthetic division in which the divisor polynomial is a monic linear polynomial. Confusingly, Ruffini's rule is sometimes referred to as synthetic division, thus leading to the common misconception that the scope of synthetic division is significantly smaller than that of the long division algorithm.For an example of Ruffini's rule, consider divided by . First, if a power of is missing from the dividend, a term with that power and a zero coefficient must be inserted into the correct position..

### Monomial order

A well ordered set of monomials which also satisfies the condition " implies " for all monomials , , and . Examples of monomial orders are the lexicographic order and the total degree order.

### Content

There are several meanings of the word content in mathematics.The content of a polytope or other -dimensional object is its generalized volume (i.e., its "hypervolume"). Just as a three-dimensional object has volume, surface area, and generalized diameter, an -dimensional object has "measures" of order 1, 2, ..., . The content of a region can be computed in the Wolfram Language using RegionMeasure[reg].The content of an integer polynomial , denoted , is the largest integer such that also has integer coefficients. Gauss's lemma for contents states that if and are two polynomials with integer coefficients, then (Séroul 2000, p. 287).For a general univariate polynomial , the Wolfram Language command FactorTermsList[poly, x] returns a list of three elements, the first being the integer content , the second being the polynomial content, i.e., a primitive (with respect to all variables) polynomial that..

### Monomial

A monomial is a product of positive integer powers of a fixed set of variables (possibly) together with a coefficient, e.g., , , or . A monomial can also be thought of as a nonzero summand of a polynomial (Becker and Weispfenning 1993, p. 191; Cox et al. 1996). A monomial with the coefficient excluded is usually called a term.Unfortunately, in some older works, the definitions of monomial and term are sometimes reversed. Care is therefore needed in attempting to distinguish these conflicting usages.The Wolfram Language command MonomialList[poly, , , ...] gives the list of monomials with respect to the variables in the specified polynomial.The monomials and are orthogonal on the unit circle in the complex plane (Dumitriu et al. 2004) since(1)The monomial functions are defined as(2)where is the set of permutations giving distinct terms in the sum and is considered to be infinite (Dumitriu et al. 2004). For example.(3)Care is needed when..

### Constant polynomial

A polynomial that, when evaluated over each in the domain of definition, results in the same value. The simplest example is for and a constant.

### Resultant

Given a polynomial(1)of degree with roots , , ..., and a polynomial(2)of degree with roots , , ..., , the resultant , also denoted and also called the eliminant, is defined by(3)(Trott 2006, p. 26).Amazingly, the resultant is also given by the determinantof the corresponding Sylvester matrix.Kronecker gave a series of lectures on resultants during the summer of 1885 (O'Connor and Robertson 2005).An important application of the resultant is the elimination of one variable from a system of two polynomial equations (Trott 2006, p. 26).The resultant of two polynomials can be computed using the Wolfram Language function Resultant[poly1, poly2, var]. This command accepts the following methods: Automatic, SylvesterMatrix, BezoutMatrix, Subresultants, and Modular, where the optimal choice depends dramatically on the concrete polynomial pair under consideration and typically requires some experimentation. For high-order..

### Irreducible polynomial

A polynomial is said to be irreducible if it cannotbe factored into nontrivial polynomials over the same field.For example, in the field of rational polynomials (i.e., polynomials with rational coefficients), is said to be irreducible if there do not exist two nonconstant polynomials and in with rational coefficients such that(Nagell 1951, p. 160). Similarly, in the finite field GF(2), is irreducible, but is not, since (mod 2).Irreducible polynomial checking is implemented in the WolframLanguage as IrreduciblePolynomialQ[poly].In general, the number of irreducible polynomials of degree over the finite field GF() is given bywhere is the Möbius function.The number of irreducible polynomials of degree over GF(2) is equal to the number of -bead fixed aperiodic necklaces of two colors and the number of binary Lyndon words of length . The first few numbers of irreducible polynomial (mod 2) for , 2, ... are 2, 1, 2, 3, 6, 9, 18, .....