Inmathematics, especially in the field ofalgebra, apolynomial ring orpolynomial algebra is aring formed from theset ofpolynomials in one or moreindeterminates (traditionally also calledvariables) withcoefficients in anotherring, often afield.^[1]

Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of theintegers.^[2]

Polynomial rings occur and are often fundamental in many parts of mathematics such asnumber theory,commutative algebra, andalgebraic geometry. Inring theory, many classes of rings, such asunique factorization domains,regular rings,group rings,rings of formal power series,Ore polynomials,graded rings, have been introduced for generalizing some properties of polynomial rings.^[3]

A closely related notion is that of thering of polynomial functions on avector space, and, more generally,ring of regular functions on analgebraic variety.^[2]

LetK be afield or (more generally) acommutative ring.

Thepolynomial ring inX overK, which is denotedK[X], can be defined in several equivalent ways. One of them is to defineK[X] as the set of expressions, calledpolynomials inX, of the form^[4]

${\displaystyle p=p_0+p_{1}X+p_2{X}^2+\cdots +p_{m-1}{X}^{m-1}+p_m{X}^m,}$

wherep₀,p₁, …,p_m, thecoefficients ofp, are elements ofK,p_m ≠ 0 ifm > 0, andX,X², …, are symbols, which are considered as "powers" ofX, and follow the usual rules ofexponentiation:X⁰ = 1,X¹ =X, and${\displaystyle X^k{X}^l=X^{k+l}}$ for anynonnegative integersk andl. The symbolX is called an indeterminate^[5] or variable.^[6] (The term of "variable" comes from the terminology ofpolynomial functions. However, here,X has no value (other than itself), and cannot vary, being aconstant in the polynomial ring.)

Two polynomials are equal when the corresponding coefficients of eachX^k are equal.

One can think of the ringK[X] as arising fromK by adding one new elementX that is external toK, commutes with all elements ofK, and has no other specific properties. This can be used for an equivalent definition of polynomial rings.

The polynomial ring inX overK is equipped with an addition, a multiplication and ascalar multiplication that make it acommutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if

${\displaystyle p=p_0+p_{1}X+p_2{X}^2+\cdots +p_m{X}^m,}$

and

${\displaystyle q=q_0+q_{1}X+q_2{X}^2+\cdots +q_n{X}^n,}$

then

${\displaystyle p+q=r_0+r_{1}X+r_2{X}^2+\cdots +r_k{X}^k,}$

and

${\displaystyle pq=s_0+s_{1}X+s_2{X}^2+\cdots +s_l{X}^l,}$

wherek = max(m,n),l =m +n,

${\displaystyle r_i=p_i+q_i}$

and

${\displaystyle s_i=p_0{q}_i+p_1{q}_{i-1}+\cdots +p_i{q}_0.}$

In these formulas, the polynomialsp andq are extended by adding "dummy terms" with zero coefficients, so that allp_i andq_i that appear in the formulas are defined. Specifically, ifm <n, thenp_i = 0 form <i ≤n.

The scalar multiplication is the special case of the multiplication wherep =p₀ is reduced to itsconstant term (the term that is independent ofX); that is

${\displaystyle p_0\left(q_0+q_{1}X+\cdots +q_n{X}^n\right)=p_0{q}_0+\left(p_0{q}_1\right)X+\cdots +\left(p_0{q}_n\right)X^n}$

It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra overK. Therefore, polynomial rings are also calledpolynomial algebras.

Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinitesequence(p₀,p₁,p₂, …) of elements ofK, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is somem so thatp_n = 0 forn >m. In this case,p₀ andX are considered as alternate notations for the sequences(p₀, 0, 0, …) and(0, 1, 0, 0, …), respectively. A straightforward use of the operation rules shows that the expression

${\displaystyle p_0+p_{1}X+p_2{X}^2+\cdots +p_m{X}^m}$

is then an alternate notation for the sequence

(p₀,p₁,p₂, …,p_m, 0, 0, …).

Let

${\displaystyle p=p_0+p_{1}X+p_2{X}^2+\cdots +p_{m-1}{X}^{m-1}+p_m{X}^m,}$

be a nonzero polynomial with${\displaystyle p_m\ne 0}$

Theconstant term ofp is${\displaystyle p_0.}$ It is zero in the case of the zero polynomial.

Thedegree ofp, writtendeg(p) is${\displaystyle m,}$ the largestk such that the coefficient ofX^k is not zero.^[7]

Theleading coefficient ofp is${\displaystyle p_m.}$^[8]

In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined,^[9] defined to be−1,^[10] or defined to be a−∞.^[11]

Aconstant polynomial is either the zero polynomial, or a polynomial of degree zero.

A nonzero polynomial ismonic if its leading coefficient is${\displaystyle 1.}$

Given two polynomialsp andq, if the degree of the zero polynomial is defined to be${\displaystyle -\mathrm{\infty },}$ one has

${\displaystyle \mathrm{deg}(p+q)\le max(\mathrm{deg}(p),\mathrm{deg}(q)),}$

and, over afield, or more generally anintegral domain,^[12]

${\displaystyle \mathrm{deg}(pq)=\mathrm{deg}(p)+\mathrm{deg}(q).}$

It follows immediately that, ifK is an integral domain, then so isK[X].^[13]

It follows also that, ifK is an integral domain, a polynomial is aunit (that is, it has amultiplicative inverse) if and only if it is constant and is a unit inK.

Two polynomials areassociated if either one is the product of the other by a unit.

Over a field, every nonzero polynomial is associated to a unique monic polynomial.

Given two polynomials,p andq, one says thatpdividesq,p is adivisor ofq, orq is a multiple ofp, if there is a polynomialr such thatq =pr.

A polynomial isirreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.

LetK be a field or, more generally, acommutative ring, andR a ring containingK. For any polynomialP inK[X] and any elementa inR, the substitution ofX witha inP defines an element ofR, which isdenotedP(a). This element is obtained by carrying on inR after the substitution the operations indicated by the expression of the polynomial. This computation is called theevaluation ofP ata. For example, if we have

${\displaystyle P=X^2-1,}$

we have

(in the first exampleR =K, and in the second oneR =K[X]). SubstitutingX for itself results in

${\displaystyle P=P(X),}$

explaining why the sentences "LetP be a polynomial" and "LetP(X) be a polynomial" are equivalent.

Thepolynomial function defined by a polynomialP is the function fromK intoK that is defined by${\displaystyle x\mapsto P(x).}$ IfK is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, ifK is a field withq elements, then the polynomials0 andX^q −X both define the zero function.

For everya inR, the evaluation ata, that is, the map${\displaystyle P\mapsto P(a)}$ defines analgebra homomorphism fromK[X] toR, which is the unique homomorphism fromK[X] toR that fixesK, and mapsX toa. In other words,K[X] has the followinguniversal property:

For every ringR containingK, and every elementa ofR, there is a unique algebra homomorphism fromK[X] toR that fixesK, and mapsX toa.

As for all universal properties, this defines the pair(K[X],X) up to a unique isomorphism, and can therefore be taken as a definition ofK[X].

Theimage of the map${\displaystyle P\mapsto P(a)}$, that is, the subset ofR obtained by substitutinga forX in elements ofK[X], is denotedK[a].^[14] For example,${\displaystyle \mathbb{Z}[\sqrt{2}]=\{P(\sqrt{2})\mid P(X)\in \mathbb{Z}[X]\}}$, and the simplification rules for the powers of a square root imply${\displaystyle \mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a\in \mathbb{Z},b\in \mathbb{Z}\}.}$

IfK is afield, the polynomial ringK[X] has many properties that are similar to those of thering of integers${\displaystyle \mathbb{Z}.}$ Most of these similarities result from the similarity between thelong division of integers and thelong division of polynomials.

Most of the properties ofK[X] that are listed in this section do not remain true ifK is not a field, or if one considers polynomials in several indeterminates.

Like for integers, theEuclidean division of polynomials has a property of uniqueness. That is, given two polynomialsa andb ≠ 0 inK[X], there is a unique pair(q,r) of polynomials such thata =bq +r, and eitherr = 0 ordeg(r) < deg(b). This makesK[X] aEuclidean domain. However, most other Euclidean domains (except integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division.

The Euclidean division is the basis of theEuclidean algorithm for polynomials that computes apolynomial greatest common divisor of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for thepreorder defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors ofa andb are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to1).

Theextended Euclidean algorithm allows computing (and proving)Bézout's identity. In the case ofK[X], it may be stated as follows. Given two polynomialsp andq of respective degreesm andn, if their monic greatest common divisorg has the degreed, then there is a unique pair(a,b) of polynomials such that

${\displaystyle ap+bq=g,}$

and

(For making this true in the limiting case wherem =d orn =d, one has to define as negative the degree of the zero polynomial. Moreover, the equality${\displaystyle \mathrm{deg}(a)=n-d}$ can occur only ifp andq are associated.) The uniqueness property is rather specific toK[X]. In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must requirea > 0.

Euclid's lemma applies toK[X]. That is, ifa dividesbc, and iscoprime withb, thena dividesc. Here,coprime means that the monic greatest common divisor is1.Proof: By hypothesis and Bézout's identity, there aree,p, andq such thatae =bc and1 =ap +bq. So${\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}$

Theunique factorization property results from Euclid's lemma. In the case of integers, this is thefundamental theorem of arithmetic. In the case ofK[X], it may be stated as:every non-constant polynomial can be expressed in a unique way as the product of a constant, and one or several irreducible monic polynomials; this decomposition is unique up to the order of the factors. In other termsK[X] is aunique factorization domain. IfK is the field of complex numbers, thefundamental theorem of algebra asserts that a univariate polynomial is irreducible if and only if its degree is one. In this case the unique factorization property can be restated as:every non-constant univariate polynomial over the complex numbers can be expressed in a unique way as the product of a constant, and one or several polynomials of the formX −r;this decomposition is unique up to the order of the factors. For each factor,r is aroot of the polynomial, and the number of occurrences of a factor is themultiplicity of the corresponding root.

The(formal) derivative of the polynomial

${\displaystyle a_0+a_{1}X+a_2{X}^2+\cdots +a_n{X}^n}$

is the polynomial

${\displaystyle a_1+2a_{2}X+\cdots +n{a}_n{X}^{n-1}.}$

In the case of polynomials withreal orcomplex coefficients, this is the standardderivative. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion oflimit is defined. The derivative makes the polynomial ring adifferential algebra.

The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers.

A polynomial with coefficients in a field or integral domain issquare-free if it does not have amultiple root in thealgebraically closed field containing its coefficients. In particular, a polynomial of degreen with real or complex coefficients is square-free if it hasn distinct complex roots. Equivalently, a polynomial over a field is square-free if and only if thegreatest common divisor of the polynomial and its derivative is1.

Asquare-free factorization of a polynomial is an expression for that polynomial as a product of powers ofpairwise relatively prime square-free factors. Over the real numbers (or any other field ofcharacteristic 0), such a factorization can be computed efficiently byYun's algorithm. Less efficient algorithms are known forsquare-free factorization of polynomials over finite fields.

Given a finite set of ordered pairs${\displaystyle (x_j,y_j)}$ with entries in a field and distinct values${\displaystyle x_j}$, among the polynomials${\displaystyle f(x)}$ that interpolate these points (so that${\displaystyle f(x_j)=y_j}$ for all${\displaystyle j}$), there is a unique polynomial of smallest degree. This is theLagrange interpolation polynomial${\displaystyle L(x)}$. If there are${\displaystyle k}$ ordered pairs, the degree of${\displaystyle L(x)}$ is at most${\displaystyle k-1}$. The polynomial${\displaystyle L(x)}$ can be computed explicitly in terms of the input data${\displaystyle (x_j,y_j)}$.

Adecomposition of a polynomial is a way of expressing it as acomposition of other polynomials of degree larger than 1. A polynomial that cannot be decomposed isindecomposable.Ritt's polynomial decomposition theorem asserts that if${\displaystyle f=g_1\circ g_2\circ \cdots \circ g_m=h_1\circ h_2\circ \cdots \circ h_n}$ are two different decompositions of the polynomial${\displaystyle f}$, then${\displaystyle m=n}$ and the degrees of the indecomposables in one decomposition are the same as the degrees of the indecomposables in the other decomposition (though not necessarily in the same order).

Except for factorization, all previous properties ofK[X] areeffective, since their proofs, as sketched above, are associated withalgorithms for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as theircomputational complexity is aquadratic function of the input size.

The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is no known algorithm running on a classical (non-quantum) computer for factorizing them inpolynomial time. This is the basis of theRSA cryptosystem, widely used for secure Internet communications.

In the case ofK[X], the factors, and the methods for computing them, depend strongly onK. Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over therational numbers, there are irreducible polynomials of any degree. For example, the polynomial${\displaystyle X^4-2}$ is irreducible over the rational numbers, is factored as${\displaystyle (X-\sqrt[4]{2})(X+\sqrt[4]{2})(X^2+\sqrt{2})}$ over the real numbers and, and as${\displaystyle (X-\sqrt[4]{2})(X+\sqrt[4]{2})(X-i\sqrt[4]{2})(X+i\sqrt[4]{2})}$ over the complex numbers.

The existence of a factorization algorithm depends also on the ground field. In the case of the real or complex numbers,Abel–Ruffini theorem shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly. Therefore, a factorization algorithm can compute only approximations of the factors. Various algorithms have been designed for computing such approximations, seeRoot finding of polynomials.

There is an example of a fieldK such that there exist exact algorithms for the arithmetic operations ofK, but there cannot exist any algorithm for deciding whether a polynomial of the form${\displaystyle X^p-a}$ isirreducible or is a product of polynomials of lower degree.^[15]

On the other hand, over the rational numbers and over finite fields, the situation is better than forinteger factorization, as there arefactorization algorithms that have apolynomial complexity. They are implemented in most general purposecomputer algebra systems.

Ifθ is an element of anassociativeK-algebraL, thepolynomial evaluation atθ is the uniquealgebra homomorphismφ fromK[X] intoL that mapsX toθ and does not affect the elements ofK itself (it is theidentity map onK). It consists ofsubstitutingX withθ in every polynomial. That is,

${\displaystyle \phi \left(a_m{X}^m+a_{m-1}{X}^{m-1}+\cdots +a_{1}X+a_0\right)=a_m{\theta }^m+a_{m-1}{\theta }^{m-1}+\cdots +a_1\theta +a_0.}$

The image of thisevaluation homomorphism is the subalgebra generated byθ, which is necessarily commutative.Ifφ is injective, the subalgebra generated byθ is isomorphic toK[X]. In this case, this subalgebra is often denoted byK[θ]. The notation ambiguity is generally harmless, because of the isomorphism.

If the evaluation homomorphism is not injective, this means that itskernel is a nonzeroideal, consisting of all polynomials that become zero whenX is substituted withθ. This ideal consists of all multiples of some monic polynomial, that is called theminimal polynomial ofθ. The termminimal is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal.

There are two main cases where minimal polynomials are considered.

Infield theory andnumber theory, an elementθ of anextension fieldL ofK isalgebraic overK if it is a root of some polynomial with coefficients inK. Theminimal polynomial overK ofθ is thus the monic polynomial of minimal degree that hasθ as a root. BecauseL is a field, this minimal polynomial is necessarilyirreducible overK. For example, the minimal polynomial (over the reals as well as over the rationals) of thecomplex numberi is${\displaystyle X^2+1}$. Thecyclotomic polynomials are the minimal polynomials of theroots of unity.

Inlinear algebra, then×nsquare matrices overK form anassociativeK-algebra of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has aminimal polynomial (not necessarily irreducible). ByCayley–Hamilton theorem, the evaluation homomorphism maps to zero thecharacteristic polynomial of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at mostn.

In the case ofK[X], thequotient ring by an ideal can be built, as in the general case, as a set ofequivalence classes. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.

Given a polynomialp of degreed, thequotient ring ofK[X] by theideal generated byp can be identified with thevector space of the polynomials of degrees less thand, with the "multiplication modulop" as a multiplication, themultiplication modulop consisting of the remainder under the division byp of the (usual) product of polynomials. This quotient ring is variously denoted as${\displaystyle K[X]/pK[X],}$${\displaystyle K[X]/⟨p⟩,}$${\displaystyle K[X]/(p),}$ or simply${\displaystyle K[X]/p.}$

The ring${\displaystyle K[X]/(p)}$ is a field if and only ifp is anirreducible polynomial. In fact, ifp is irreducible, every nonzero polynomialq of lower degree is coprime withp, andBézout's identity allows computingr ands such thatsp +qr = 1; so,r is themultiplicative inverse ofq modulop. Conversely, ifp is reducible, then there exist polynomialsa, b of degrees lower thandeg(p) such thatab =p ; soa, b are nonzerozero divisors modulop, and cannot be invertible.

For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring

${\displaystyle \mathbb{C}=\mathbb{R}[X]/(X^2+1),}$

and that the image ofX in${\displaystyle \mathbb{C}}$ is denoted byi. In fact, by the above description, this quotient consists of all polynomials of degree one ini, which have the forma +bi, witha andb in${\displaystyle \mathbb{R}.}$ The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacingi² by−1 in their product as polynomials (this is exactly the usual definition of the product of complex numbers).

Letθ be analgebraic element in aK-algebraA. Byalgebraic, one means thatθ has a minimal polynomialp. Thefirst ring isomorphism theorem asserts that the substitution homomorphism induces anisomorphism of${\displaystyle K[X]/(p)}$ onto the imageK[θ] of the substitution homomorphism. In particular, ifA is asimple extension ofK generated byθ, this allows identifyingA and${\displaystyle K[X]/(p).}$ This identification is widely used inalgebraic number theory.

Thestructure theorem for finitely generated modules over a principal ideal domain applies toK[X], whenK is a field. This means that every finitely generated module overK[X] may be decomposed into adirect sum of afree module and finitely many modules of the form${\displaystyle K[X]/⟨P^k⟩}$, whereP is anirreducible polynomial overK andk a positive integer.

Givenn symbols${\displaystyle X_1,\dots ,X_n,}$ calledindeterminates, amonomial (also calledpower product)

${\displaystyle X_1^{{\alpha }_1}\cdots X_n^{{\alpha }_n}}$

is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular,${\displaystyle X_1^0\cdots X_n^0=1.}$

Thetuple of exponentsα = (α₁, …,α_n) is called themultidegree orexponent vector of the monomial. For a less cumbersome notation, the abbreviation

${\displaystyle X^{\alpha }=X_1^{{\alpha }_1}\cdots X_n^{{\alpha }_n}}$

is often used. Thedegree of a monomialX^α, frequently denoteddegα or|α|, is the sum of its exponents:

${\displaystyle \mathrm{deg}\alpha =\sum _{i=1}^n{\alpha }_i.}$

Apolynomial in these indeterminates, with coefficients in a fieldK, or more generally aring, is a finitelinear combination of monomials

${\displaystyle p=\sum _{\alpha }{p}_{\alpha }{X}^{\alpha }}$

with coefficients inK. Thedegree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.

The set of polynomials in${\displaystyle X_1,\dots ,X_n,}$ denoted${\displaystyle K[X_1,\dots ,X_n],}$ is thus avector space (or afree module, ifK is a ring) that has the monomials as a basis.

${\displaystyle K[X_1,\dots ,X_n]}$ is naturally equipped (see below) with a multiplication that makes aring, and anassociative algebra overK, calledthe polynomial ring inn indeterminates overK (the definite articlethe reflects that it is uniquely defined up to the name and the order of the indeterminates. If the ringK iscommutative,${\displaystyle K[X_1,\dots ,X_n]}$ is also a commutative ring.

Addition andscalar multiplication of polynomials are those of avector space orfree module equipped by a specific basis (here the basis of the monomials). Explicitly, let${\displaystyle p=\sum _{\alpha \in I}{p}_{\alpha }{X}^{\alpha },q=\sum _{\beta \in J}{q}_{\beta }{X}^{\beta },}$whereI andJ are finite sets of exponent vectors.

The scalar multiplication ofp and a scalar${\displaystyle c\in K}$ is

${\displaystyle cp=\sum _{\alpha \in I}c{p}_{\alpha }{X}^{\alpha }.}$

The addition ofp andq is

${\displaystyle p+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}$

where${\displaystyle p_{\alpha }=0}$ if${\displaystyle \alpha \notin I,}$ and${\displaystyle q_{\beta }=0}$ if${\displaystyle \beta \notin J.}$ Moreover, if one has${\displaystyle p_{\alpha }+q_{\alpha }=0}$ for some${\displaystyle \alpha \in I\cap J,}$ the corresponding zero term is removed from the result.

The multiplication is

${\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }{p}_{\alpha }{q}_{\beta }\right)X^{\gamma },}$

where${\displaystyle I+J}$ is the set of the sums of one exponent vector inI and one other inJ (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the factors.

The verification of the axioms of anassociative algebra is straightforward.

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Apolynomial expression is anexpression built with scalars (elements ofK), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.

As all these operations are defined in${\displaystyle K[X_1,\dots ,X_n]}$ a polynomial expression represents a polynomial, that is an element of${\displaystyle K[X_1,\dots ,X_n].}$ The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called thecanonical form,normal form, orexpanded form of the polynomial. Given a polynomial expression, one can compute theexpanded form of the represented polynomial byexpanding with thedistributive law all the products that have a sum among their factors, and then usingcommutativity (except for the product of two scalars), andassociativity for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping thelike terms.

The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise ofcomputer algebra, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.

IfK is a commutative ring, the polynomial ringK[X₁, …,X_n] has the followinguniversal property: for everycommutativeK-algebraA, and everyn-tuple(x₁, …,x_n) of elements ofA, there is a uniquealgebra homomorphism fromK[X₁, …,X_n] toA that maps each${\displaystyle X_i}$ to the corresponding${\displaystyle x_i.}$ This homomorphism is theevaluation homomorphism that consists in substituting${\displaystyle X_i}$ with${\displaystyle x_i}$ in every polynomial.

As it is the case for every universal property, this characterizes the pair${\displaystyle (K[X_1,\dots ,X_n],(X_1,\dots ,X_n))}$ up to a uniqueisomorphism.

This may also be interpreted in terms ofadjoint functors. More precisely, letSET andALG be respectively thecategories of sets and commutativeK-algebras (here, and in the following, the morphisms are trivially defined). There is aforgetful functor${\displaystyle \mathrm{F}:\mathrm{A}\mathrm{L}\mathrm{G}\to \mathrm{S}\mathrm{E}\mathrm{T}}$ that maps algebras to their underlying sets. On the other hand, the map${\displaystyle X\mapsto K[X]}$ defines a functor${\displaystyle \mathrm{P}\mathrm{O}\mathrm{L}:\mathrm{S}\mathrm{E}\mathrm{T}\to \mathrm{A}\mathrm{L}\mathrm{G}}$ in the other direction. (IfX is infinite,K[X] is the set of all polynomials in a finite number of elements ofX.)

The universal property of the polynomial ring means thatF andPOL areadjoint functors. That is, there is a bijection

${\displaystyle {\mathrm{Hom}}_{\mathrm{S}\mathrm{E}\mathrm{T}}(X,\mathrm{F}(A))\cong {\mathrm{Hom}}_{\mathrm{A}\mathrm{L}\mathrm{G}}(K[X],A).}$

This may be expressed also by saying that polynomial rings arefree commutative algebras, since they arefree objects in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is thefree commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.

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A polynomial in${\displaystyle K[X_1,\dots ,X_n]}$ can be considered as a univariate polynomial in the indeterminate${\displaystyle X_n}$ over the ring${\displaystyle K[X_1,\dots ,X_{n-1}],}$ by regrouping the terms that contain the same power of${\displaystyle X_n,}$ that is, by using the identity

${\displaystyle \sum _{({\alpha }_1,\dots ,{\alpha }_n)\in I}{c}_{{\alpha }_1,\dots ,{\alpha }_n}{X}_1^{{\alpha }_1}\cdots X_n^{{\alpha }_n}=\sum _i\left(\sum _{({\alpha }_1,\dots ,{\alpha }_{n-1})\mid ({\alpha }_1,\dots ,{\alpha }_{n-1},i)\in I}{c}_{{\alpha }_1,\dots ,{\alpha }_{n-1}}{X}_1^{{\alpha }_1}\cdots X_{n-1}^{{\alpha }_{n-1}}\right)X_n^i,}$

which results from the distributivity and associativity of ring operations.

This means that one has analgebra isomorphism

${\displaystyle K[X_1,\dots ,X_n]\cong (K[X_1,\dots ,X_{n-1}])[X_n]}$

that maps each indeterminate to itself. (This isomorphism is often written as an equality, which is justified by the fact that polynomial rings are defined up to aunique isomorphism.)

In other words, a multivariate polynomial ring can be considered as a univariate polynomial over a smaller polynomial ring. This is commonly used for proving properties of multivariate polynomial rings, byinduction on the number of indeterminates.

The main such properties are listed below.

In this section,R is a commutative ring,K is a field,X denotes a single indeterminate, and, as usual,${\displaystyle \mathbb{Z}}$ is the ring of integers. Here is the list of the main ring properties that remain true when passing fromR toR[X].

• IfR is anintegral domain then the same holds forR[X] (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors).
  • In particular,${\displaystyle K[X_1,\dots ,X_n]}$ and${\displaystyle \mathbb{Z}[X_1,\dots ,X_n]}$ are integral domains.
• IfR is aunique factorization domain then the same holds forR[X]. This results fromGauss's lemma and the unique factorization property of${\displaystyle L[X],}$ whereL is the field of fractions ofR.
  • In particular,${\displaystyle K[X_1,\dots ,X_n]}$ and${\displaystyle \mathbb{Z}[X_1,\dots ,X_n]}$ are unique factorization domains.
• IfR is aNoetherian ring, then the same holds forR[X].
  • In particular,${\displaystyle K[X_1,\dots ,X_n]}$ and${\displaystyle \mathbb{Z}[X_1,\dots ,X_n]}$ are Noetherian rings; this isHilbert's basis theorem.
• IfR is a Noetherian ring, then${\displaystyle \dim R[X]=1+\dim R,}$ where "${\displaystyle \dim }$" denotes theKrull dimension.
  • In particular,${\displaystyle \dim K[X_1,\dots ,X_n]=n}$ and${\displaystyle \dim \mathbb{Z}[X_1,\dots ,X_n]=n+1.}$
• IfR is aregular ring, then the same holds forR[X]; in this case, one has$${\displaystyle \mathrm{gl}\dim R[X]=\dim R[X]=1+\mathrm{gl}\dim R=1+\dim R,}$$ where "${\displaystyle \mathrm{gl}\dim }$" denotes theglobal dimension.
  • In particular,${\displaystyle K[X_1,\dots ,X_n]}$ and${\displaystyle \mathbb{Z}[X_1,\dots ,X_n]}$ are regular rings,${\displaystyle \mathrm{gl}\dim \mathbb{Z}[X_1,\dots ,X_n]=n+1,}$ and${\displaystyle \mathrm{gl}\dim K[X_1,\dots ,X_n]=n.}$ The latter equality isHilbert's syzygy theorem.

Polynomial rings in several variables over a field are fundamental ininvariant theory andalgebraic geometry. Some of their properties, such as those described above can be reduced to the case of a single indeterminate, but this is not always the case. In particular, because of the geometric applications, many interesting properties must be invariant underaffine orprojective transformations of the indeterminates. This often implies that one cannot select one of the indeterminates for a recurrence on the indeterminates.

Bézout's theorem,Hilbert's Nullstellensatz andJacobian conjecture are among the most famous properties that are specific to multivariate polynomials over a field.

The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved byDavid Hilbert, which extends to the multivariate case some aspects of thefundamental theorem of algebra. It is foundational foralgebraic geometry, as establishing a strong link between the algebraic properties of${\displaystyle K[X_1,\dots ,X_n]}$ and the geometric properties ofalgebraic varieties, that are (roughly speaking) set of points defined byimplicit polynomial equations.

The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz.

The first version generalizes the fact that a nonzero univariate polynomial has acomplex zero if and only if it is not a constant. The statement is:a set of polynomialsS in${\displaystyle K[X_1,\dots ,X_n]}$ has a common zero in analgebraically closed field containingK, if and only if1does not belong to theideal generated byS, that is, if1is not alinear combination of elements ofS with polynomial coefficients.

The second version generalizes the fact that theirreducible univariate polynomials over the complex numbers areassociate to a polynomial of the form${\displaystyle X-\alpha .}$ The statement is:IfK is algebraically closed, then themaximal ideals of${\displaystyle K[X_1,\dots ,X_n]}$ have the form${\displaystyle ⟨X_1-{\alpha }_1,\dots ,X_n-{\alpha }_n⟩.}$

Bézout's theorem may be viewed as a multivariate generalization of the version of thefundamental theorem of algebra that asserts that a univariate polynomial of degreen hasn complex roots, if they are counted with their multiplicities.

In the case ofbivariate polynomials, it states that two polynomials of degreesd ande in two variables, which have no common factors of positive degree, have exactlyde common zeros in analgebraically closed field containing the coefficients, if the zeros are counted with their multiplicity and include thezeros at infinity.

For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work withhomogeneous polynomials, and consider zeros in aprojective space. In this context, aprojective zero of a homogeneous polynomial${\displaystyle P(X_0,\dots ,X_n)}$ is, up to a scaling, a(n + 1)-tuple${\displaystyle (x_0,\dots ,x_n)}$ of elements ofK that is different from(0, …, 0), and such that${\displaystyle P(x_0,\dots ,x_n)=0}$. Here, "up to a scaling" means that${\displaystyle (x_0,\dots ,x_n)}$ and${\displaystyle (\lambda x_0,\dots ,\lambda x_n)}$ are considered as the same zero for any nonzero${\displaystyle \lambda \in K.}$ In other words, a zero is a set ofhomogeneous coordinates of a point in a projective space of dimensionn.

Then, Bézout's theorem states: Givenn homogeneous polynomials of degrees${\displaystyle d_1,\dots ,d_n}$ inn + 1 indeterminates, which have only a finite number of common projective zeros in analgebraically closed extension ofK, the sum of themultiplicities of these zeros is the product${\displaystyle d_1\cdots d_n.}$

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Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings,noncommutative polynomial rings,skew polynomial rings, and polynomialrigs.

One slight generalization of polynomial rings is to allow for infinitely many indeterminates. Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials. Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being thefree commutative algebra, the only difference is that it is afree object over an infinite set.

One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than thering of power series in infinitely many variables. Such a ring is used for constructing thering of symmetric functions over an infinite set.

A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents:Xⁱ ⋅X^j =X^i+j. A set for which addition makes sense (is closed and associative) is called amonoid. The set of functions from a monoidN to a ringR which are nonzero at only finitely many places can be given the structure of a ring known asR[N], themonoid ring ofN with coefficients inR. The addition is defined component-wise, so that ifc =a +b, thenc_n =a_n +b_n for everyn inN. The multiplication is defined as the Cauchy product, so that ifc =a ⋅b, then for eachn inN,c_n is the sum of alla_ib_j wherei,j range over all pairs of elements ofN which sum ton.

WhenN is commutative, it is convenient to denote the functiona inR[N] as the formal sum:

${\displaystyle \sum _{n\in N}{a}_n{X}^n}$

and then the formulas for addition and multiplication are the familiar:

${\displaystyle \left(\sum _{n\in N}{a}_n{X}^n\right)+\left(\sum _{n\in N}{b}_n{X}^n\right)=\sum _{n\in N}\left(a_n+b_n\right)X^n}$

and

${\displaystyle \left(\sum _{n\in N}{a}_n{X}^n\right)\cdot \left(\sum _{n\in N}{b}_n{X}^n\right)=\sum _{n\in N}\left(\sum _{i+j=n}{a}_i{b}_j\right)X^n}$

where the latter sum is taken over alli,j inN that sum ton.

Some authors such as (Lang 2002, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case whereN is the monoid of non-negative integers. Polynomials in several variables simply takeN to be the direct product of several copies of the monoid of non-negative integers.

Several interesting examples of rings and groups are formed by takingN to be the additive monoid of non-negative rational numbers, (Osborne 2000, §4.4). See alsoPuiseux series.

Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoidN used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice ofN, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions fromN to a ringR with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as thering completion of the polynomial ring with respect to the ideal generated byx.

For polynomial rings of more than one variable, the productsX⋅Y andY⋅X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring inn noncommuting variables with coefficients in the ringR is themonoid ringR[N], where the monoidN is thefree monoid onn letters, also known as the set of all strings over an alphabet ofn symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.

Just as the polynomial ring inn variables with coefficients in the commutative ringR is the free commutativeR-algebra of rankn, the noncommutative polynomial ring inn variables with coefficients in the commutative ringR is the free associative, unitalR-algebra onn generators, which is noncommutative whenn > 1.

Other generalizations of polynomials are differential and skew-polynomial rings.

Adifferential polynomial ring is a ring ofdifferential operators formed from a ringR and aderivationδ ofR intoR. This derivation operates onR, and will be denotedX, when viewed as an operator. The elements ofR also operate onR by multiplication. Thecomposition of operators is denoted as the usual multiplication. It follows that the relationδ(ab) =aδ(b) +δ(a)b may be rewrittenas

${\displaystyle X\cdot a=a\cdot X+\delta (a).}$

This relation may be extended to define a skew multiplication between two polynomials inX with coefficients inR, which make them anoncommutative ring.

The standard example, called aWeyl algebra, takesR to be a (usual) polynomial ringk[Y ], andδ to be the standard polynomial derivative${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}$. Takinga =Y in the above relation, one gets thecanonical commutation relation,X⋅Y −Y⋅X = 1. Extending this relation by associativity and distributivity allows explicitly constructing theWeyl algebra. (Lam 2001, §1,ex1.9).

Theskew-polynomial ring is defined similarly for a ringR and a ringendomorphismf ofR, by extending the multiplication from the relationX⋅r =f(r)⋅X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphismF from the monoidN of the positive integers into theendomorphism ring ofR, the formulaXⁿ⋅r =F(n)(r)⋅Xⁿ allows constructing a skew-polynomial ring. (Lam 2001, §1,ex 1.11) Skew polynomial rings are closely related tocrossed product algebras.

The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structureR be afield or aring to the requirement thatR only be asemifield orrig; the resulting polynomial structure/extensionR[X] is apolynomial rig. For example, the set of all multivariate polynomials withnatural number coefficients is a polynomial rig.

• Additive polynomial
• Laurent polynomial

• Hall, F. M. (1969), "Section 3.6",An Introduction to Abstract Algebra, vol. 2, Cambridge University Press,ISBN 0521084849
• Herstein, I. N. (1975), "Section 3.9",Topics in Algebra, Wiley,ISBN 0471010901,polynomial ring.
• Lam, Tsit-Yuen (2001),A First Course in Noncommutative Rings,Springer-Verlag,ISBN 978-0-387-95325-0
• Lang, Serge (2002),Algebra,Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4,MR 1878556
• Osborne, M. Scott (2000),Basic homological algebra, Graduate Texts in Mathematics, vol. 196,Springer-Verlag,doi:10.1007/978-1-4612-1278-2,ISBN 978-0-387-98934-1,MR 1757274

• Commutative algebra
• Invariant theory
• Ring theory
• Polynomials
• Free algebraic structures

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