Template:Short descriptionTemplate:Ring theory sidebar Template:CS1 config In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
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 the integers.
Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique 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.
A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety.
Definition (univariate case)Edit
Let Template:Math be a field or (more generally) a commutative ring.
The polynomial ring in Template:Math over Template:Math, which is denoted Template:Math, can be defined in several equivalent ways. One of them is to define Template:Math as the set of expressions, called polynomials in Template:Math, of the form<ref>Template:Harvnb</ref>
- <math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,</math>
where Template:Math, the coefficients of Template:Math, are elements of Template:Math, Template:Math if Template:Math, and Template:Math are symbols, which are considered as "powers" of Template:Math, and follow the usual rules of exponentiation: Template:Math, Template:Math, and <math> X^k\, X^l = X^{k+l}</math> for any nonnegative integers Template:Math and Template:Math. The symbol Template:Math is called an indeterminate<ref>Herstein, Hall p. 73</ref> or variable.<ref>Template:Harvnb</ref> (The term of "variable" comes from the terminology of polynomial functions. However, here, Template:Mvar has no value (other than itself), and cannot vary, being a constant in the polynomial ring.)
Two polynomials are equal when the corresponding coefficients of each Template:Math are equal.
One can think of the ring Template:Math as arising from Template:Math by adding one new element Template:Math that is external to Template:Math, commutes with all elements of Template:Math, and has no other specific properties. This can be used for an equivalent definition of polynomial rings.
The polynomial ring in Template:Math over Template:Math is equipped with an addition, a multiplication and a scalar multiplication that make it a commutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if
- <math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m,</math>
and
- <math>q = q_0 + q_1 X + q_2 X^2 + \cdots + q_n X^n,</math>
then
- <math>p + q = r_0 + r_1 X + r_2 X^2 + \cdots + r_k X^k,</math>
and
- <math>pq = s_0 + s_1 X + s_2 X^2 + \cdots + s_l X^l,</math>
where Template:Math,
- <math>r_i = p_i + q_i</math>
and
- <math>s_i = p_0 q_i + p_1 q_{i-1} + \cdots + p_i q_0.</math>
In these formulas, the polynomials Template:Math and Template:Math are extended by adding "dummy terms" with zero coefficients, so that all Template:Math and Template:Math that appear in the formulas are defined. Specifically, if Template:Math, then Template:Math for Template:Math.
The scalar multiplication is the special case of the multiplication where Template:Math is reduced to its constant term (the term that is independent of Template:Math); that is
- <math>p_0\left(q_0 + q_1 X + \dots + 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</math>
It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over Template:Mvar. Therefore, polynomial rings are also called polynomial 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 infinite sequence Template:Math of elements of Template:Math, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some Template:Math so that Template:Nowrap for Template:Math. In this case, Template:Math and Template:Mvar are considered as alternate notations for the sequences Template:Math and Template:Math, respectively. A straightforward use of the operation rules shows that the expression
- <math>p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m</math>
is then an alternate notation for the sequence
TerminologyEdit
Let
- <math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,</math>
be a nonzero polynomial with <math>p_m\ne 0</math>
The constant term of Template:Math is <math>p_0.</math> It is zero in the case of the zero polynomial.
The degree of Template:Math, written Template:Math is <math>m,</math> the largest Template:Math such that the coefficient of Template:Math is not zero.<ref>Template:Harvnb</ref>
The leading coefficient of Template:Math is <math>p_m.</math><ref>Template:Harvnb</ref>
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,<ref>Template:Citation.</ref> defined to be Template:Math,<ref>Template:Citation.</ref> or defined to be a Template:Math.<ref>Template:Citation.</ref>
A constant polynomial is either the zero polynomial, or a polynomial of degree zero.
A nonzero polynomial is monic if its leading coefficient is <math>1.</math>
Given two polynomials Template:Mvar and Template:Mvar, if the degree of the zero polynomial is defined to be <math>-\infty,</math> one has
- <math>\deg(p+q) \le \max (\deg(p), \deg (q)),</math>
and, over a field, or more generally an integral domain,<ref>Template:Harvnb</ref>
- <math>\deg(pq) = \deg(p) + \deg(q).</math>
It follows immediately that, if Template:Math is an integral domain, then so is Template:Math.<ref>Template:Harvnb</ref>
It follows also that, if Template:Math is an integral domain, a polynomial is a unit (that is, it has a multiplicative inverse) if and only if it is constant and is a unit in Template:Mvar.
Two polynomials are associated 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, Template:Mvar and Template:Mvar, one says that Template:Mvar divides Template:Mvar, Template:Mvar is a divisor of Template:Mvar, or Template:Mvar is a multiple of Template:Mvar, if there is a polynomial Template:Mvar such that Template:Math.
A polynomial is irreducible 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.
Polynomial evaluationEdit
Template:Further Let Template:Mvar be a field or, more generally, a commutative ring, and Template:Mvar a ring containing Template:Mvar. For any polynomial Template:Mvar in Template:Math and any element Template:Mvar in Template:Mvar, the substitution of Template:Mvar with Template:Mvar in Template:Mvar defines an element of Template:Math, which is denoted Template:Math. This element is obtained by carrying on in Template:Mvar after the substitution the operations indicated by the expression of the polynomial. This computation is called the evaluation of Template:Math at Template:Math. For example, if we have
- <math>P = X^2 - 1,</math>
we have
- <math>\begin{align}
P(3) &= 3^2-1 = 8, \\ P(X^2+1) &= \left(X^2 + 1\right)^2 - 1 = X^4 + 2X^2
\end{align}</math>
(in the first example Template:Math, and in the second one Template:Math). Substituting Template:Math for itself results in
- <math>P = P(X),</math>
explaining why the sentences "Let Template:Mvar be a polynomial" and "Let Template:Math be a polynomial" are equivalent.
The polynomial function defined by a polynomial Template:Mvar is the function from Template:Mvar into Template:Mvar that is defined by <math>x\mapsto P(x).</math> If Template:Mvar is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if Template:Mvar is a field with Template:Mvar elements, then the polynomials Template:Math and Template:Math both define the zero function.
For every Template:Math in Template:Math, the evaluation at Template:Mvar, that is, the map <math>P \mapsto P(a)</math> defines an algebra homomorphism from Template:Math to Template:Math, which is the unique homomorphism from Template:Math to Template:Math that fixes Template:Mvar, and maps Template:Mvar to Template:Mvar. In other words, Template:Math has the following universal property:
- For every ring Template:Mvar containing Template:Mvar, and every element Template:Mvar of Template:Mvar, there is a unique algebra homomorphism from Template:Math to Template:Mvar that fixes Template:Mvar, and maps Template:Mvar to Template:Mvar.
As for all universal properties, this defines the pair Template:Math up to a unique isomorphism, and can therefore be taken as a definition of Template:Math.
The image of the map <math>P \mapsto P(a)</math>, that is, the subset of Template:Mvar obtained by substituting Template:Mvar for Template:Mvar in elements of Template:Math, is denoted Template:Math.<ref>Knapp, Anthony W. (2006), Basic Algebra, Birkhäuser, p. 121.</ref> For example, <math>\Z[\sqrt{2}]=\{P(\sqrt{2})\mid P(X)\in\Z[X]\}</math>, and the simplification rules for the powers of a square root imply <math>\Z[\sqrt{2}]= \{a+b\sqrt 2 \mid a\in \Z, b\in \Z\}.</math>
Univariate polynomials over a fieldEdit
If Template:Mvar is a field, the polynomial ring Template:Math has many properties that are similar to those of the ring of integers <math>\Z.</math> Most of these similarities result from the similarity between the long division of integers and the long division of polynomials.
Most of the properties of Template:Math that are listed in this section do not remain true if Template:Mvar is not a field, or if one considers polynomials in several indeterminates.
Like for integers, the Euclidean division of polynomials has a property of uniqueness. That is, given two polynomials Template:Mvar and Template:Math in Template:Math, there is a unique pair Template:Math of polynomials such that Template:Math, and either Template:Math or Template:Math. This makes Template:Math a Euclidean 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 the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the preorder 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 of Template:Mvar and Template:Mvar are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to Template:Val).
The extended Euclidean algorithm allows computing (and proving) Bézout's identity. In the case of Template:Math, it may be stated as follows. Given two polynomials Template:Mvar and Template:Mvar of respective degrees Template:Mvar and Template:Mvar, if their monic greatest common divisor Template:Mvar has the degree Template:Mvar, then there is a unique pair Template:Math of polynomials such that
- <math>ap + bq = g,</math>
and
- <math>\deg (a) \le n-d, \quad \deg(b) < m-d.</math>
(For making this true in the limiting case where Template:Math or Template:Math, one has to define as negative the degree of the zero polynomial. Moreover, the equality <math>\deg (a)= n-d</math> can occur only if Template:Mvar and Template:Math are associated.) The uniqueness property is rather specific to Template:Math. In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require Template:Math.
Euclid's lemma applies to Template:Math. That is, if Template:Mvar divides Template:Mvar, and is coprime with Template:Mvar, then Template:Mvar divides Template:Mvar. Here, coprime means that the monic greatest common divisor is Template:Val. Proof: By hypothesis and Bézout's identity, there are Template:Mvar, Template:Mvar, and Template:Mvar such that Template:Math and Template:Math. So <math>c=c(ap+bq)=cap+aeq=a(cp+eq).</math>
The unique factorization property results from Euclid's lemma. In the case of integers, this is the fundamental theorem of arithmetic. In the case of Template:Math, 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 terms Template:Math is a unique factorization domain. If Template:Mvar is the field of complex numbers, the fundamental 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 form Template:Math; this decomposition is unique up to the order of the factors. For each factor, Template:Mvar is a root of the polynomial, and the number of occurrences of a factor is the multiplicity of the corresponding root.
DerivationEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The (formal) derivative of the polynomial
- <math>a_0+a_1X+a_2X^2+\cdots+a_nX^n</math>
is the polynomial
- <math>a_1+2a_2X+\cdots+na_nX^{n-1}.</math>
In the case of polynomials with real or complex coefficients, this is the standard derivative. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of limit is defined. The derivative makes the polynomial ring a differential 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.
Square-free factorizationEdit
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A polynomial with coefficients in a field or integral domain is square-free if it does not have a multiple root in the algebraically closed field containing its coefficients. In particular, a polynomial of degree Template:Mvar with real or complex coefficients is square-free if it has Template:Mvar distinct complex roots. Equivalently, a polynomial over a field is square-free if and only if the greatest common divisor of the polynomial and its derivative is Template:Math.
A square-free factorization of a polynomial is an expression for that polynomial as a product of powers of pairwise relatively prime square-free factors. Over the real numbers (or any other field of characteristic 0), such a factorization can be computed efficiently by Yun's algorithm. Less efficient algorithms are known for square-free factorization of polynomials over finite fields.
Lagrange interpolationEdit
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Given a finite set of ordered pairs <math>(x_j, y_j)</math> with entries in a field and distinct values <math>x_j</math>, among the polynomials <math>f(x)</math> that interpolate these points (so that <math>f(x_j) = y_j</math> for all <math>j</math>), there is a unique polynomial of smallest degree. This is the Lagrange interpolation polynomial <math>L(x)</math>. If there are <math>k</math> ordered pairs, the degree of <math>L(x)</math> is at most <math>k - 1</math>. The polynomial <math>L(x)</math> can be computed explicitly in terms of the input data <math>(x_j, y_j)</math>.
Polynomial decompositionEdit
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A decomposition of a polynomial is a way of expressing it as a composition of other polynomials of degree larger than 1. A polynomial that cannot be decomposed is indecomposable. Ritt's polynomial decomposition theorem asserts that if <math>f = g_1 \circ g_2 \circ \cdots \circ g_m = h_1 \circ h_2 \circ \cdots\circ h_n</math> are two different decompositions of the polynomial <math>f</math>, then <math>m = n</math> 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).
FactorizationEdit
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Except for factorization, all previous properties of Template:Math are effective, since their proofs, as sketched above, are associated with algorithms for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their computational complexity is a quadratic 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 in polynomial time. This is the basis of the RSA cryptosystem, widely used for secure Internet communications.
In the case of Template:Math, the factors, and the methods for computing them, depend strongly on Template:Mvar. 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 the rational numbers, there are irreducible polynomials of any degree. For example, the polynomial <math>X^4-2</math> is irreducible over the rational numbers, is factored as <math>(X - \sqrt[4]2)(X+\sqrt[4]2)(X^2+\sqrt 2)</math> over the real numbers and, and as <math>(X-\sqrt[4]2)(X+\sqrt[4]2)(X-i\sqrt[4]2)(X+i\sqrt[4]2)</math> 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, see Root finding of polynomials.
There is an example of a field Template:Math such that there exist exact algorithms for the arithmetic operations of Template:Math, but there cannot exist any algorithm for deciding whether a polynomial of the form <math>X^p - a</math> is irreducible or is a product of polynomials of lower degree.<ref>Template:Citation</ref>
On the other hand, over the rational numbers and over finite fields, the situation is better than for integer factorization, as there are factorization algorithms that have a polynomial complexity. They are implemented in most general purpose computer algebra systems.
Minimal polynomialEdit
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If Template:Math is an element of an [[associative algebra|associative Template:Mvar-algebra]] Template:Math, the polynomial evaluation at Template:Math is the unique algebra homomorphism Template:Math from Template:Math into Template:Math that maps Template:Math to Template:Math and does not affect the elements of Template:Math itself (it is the identity map on Template:Math). It consists of substituting Template:Math with Template:Math in every polynomial. That is,
- <math>
\varphi\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.
</math>
The image of this evaluation homomorphism is the subalgebra generated by Template:Mvar, which is necessarily commutative. If Template:Math is injective, the subalgebra generated by Template:Mvar is isomorphic to Template:Math. In this case, this subalgebra is often denoted by Template:Math. The notation ambiguity is generally harmless, because of the isomorphism.
Template:Anchor If the evaluation homomorphism is not injective, this means that its kernel is a nonzero ideal, consisting of all polynomials that become zero when Template:Mvar is substituted with Template:Mvar. This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of Template:Mvar. The term minimal 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.
In field theory and number theory, an element Template:Mvar of an extension field Template:Mvar of Template:Mvar is algebraic over Template:Mvar if it is a root of some polynomial with coefficients in Template:Mvar. The minimal polynomial over Template:Mvar of Template:Mvar is thus the monic polynomial of minimal degree that has Template:Mvar as a root. Because Template:Mvar is a field, this minimal polynomial is necessarily irreducible over Template:Mvar. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number Template:Mvar is <math>X^2 + 1</math>. The cyclotomic polynomials are the minimal polynomials of the roots of unity.
In linear algebra, the Template:Math square matrices over Template:Mvar form an [[associative algebra|associative Template:Mvar-algebra]] of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a minimal polynomial (not necessarily irreducible). By Cayley–Hamilton theorem, the evaluation homomorphism maps to zero the characteristic 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 most Template:Mvar.
Quotient ringEdit
In the case of Template:Math, the quotient ring by an ideal can be built, as in the general case, as a set of equivalence classes. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.
Given a polynomial Template:Mvar of degree Template:Mvar, the quotient ring of Template:Math by the ideal generated by Template:Mvar can be identified with the vector space of the polynomials of degrees less than Template:Mvar, with the "multiplication modulo Template:Mvar" as a multiplication, the multiplication modulo Template:Mvar consisting of the remainder under the division by Template:Mvar of the (usual) product of polynomials. This quotient ring is variously denoted as <math>K[X]/pK[X],</math> <math>K[X]/\langle p \rangle,</math> <math>K[X]/(p),</math> or simply <math>K[X]/p.</math>
The ring <math>K[X]/(p)</math> is a field if and only if Template:Mvar is an irreducible polynomial. In fact, if Template:Mvar is irreducible, every nonzero polynomial Template:Mvar of lower degree is coprime with Template:Mvar, and Bézout's identity allows computing Template:Mvar and Template:Mvar such that Template:Math; so, Template:Mvar is the multiplicative inverse of Template:Mvar modulo Template:Mvar. Conversely, if Template:Mvar is reducible, then there exist polynomials Template:Mvar of degrees lower than Template:Math such that Template:Math ; so Template:Mvar are nonzero zero divisors modulo Template:Mvar, 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
- <math>\mathbb C =\mathbb R[X]/(X^2+1),</math>
and that the image of Template:Mvar in <math>\mathbb C</math> is denoted by Template:Mvar. In fact, by the above description, this quotient consists of all polynomials of degree one in Template:Mvar, which have the form Template:Math, with Template:Mvar and Template:Mvar in <math>\mathbb R.</math> The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing Template:Math by Template:Math in their product as polynomials (this is exactly the usual definition of the product of complex numbers).
Let Template:Math be an algebraic element in a Template:Mvar-algebra Template:Mvar. By algebraic, one means that Template:Math has a minimal polynomial Template:Mvar. The first ring isomorphism theorem asserts that the substitution homomorphism induces an isomorphism of <math>K[X]/(p)</math> onto the image Template:Math of the substitution homomorphism. In particular, if Template:Mvar is a simple extension of Template:Mvar generated by Template:Math, this allows identifying Template:Mvar and <math>K[X]/(p).</math> This identification is widely used in algebraic number theory.
ModulesEdit
The structure theorem for finitely generated modules over a principal ideal domain applies to K[X], when K is a field. This means that every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitely many modules of the form <math>K[X]/\left\langle P^k \right\rangle</math>, where P is an irreducible polynomial over K and k a positive integer.
Definition (multivariate case)Edit
Template:Anchor Given Template:Mvar symbols <math>X_1, \dots, X_n,</math> called indeterminates, a monomial (also called power product)
- <math>X_1^{\alpha_1}\cdots X_n^{\alpha_n}</math>
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, <math>X_1^0\cdots X_n^0 =1.</math>
The tuple of exponents Template:Math is called the multidegree or exponent vector of the monomial. For a less cumbersome notation, the abbreviation
- <math>X^\alpha=X_1^{\alpha_1}\cdots X_n^{\alpha_n}</math>
is often used. The degree of a monomial Template:Math, frequently denoted Template:Math or Template:Math, is the sum of its exponents:
- <math> \deg \alpha = \sum_{i=1}^n \alpha_i. </math>
A polynomial in these indeterminates, with coefficients in a field Template:Mvar, or more generally a ring, is a finite linear combination of monomials
- <math> p = \sum_\alpha p_\alpha X^\alpha</math>
with coefficients in Template:Mvar. The degree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.
The set of polynomials in <math>X_1, \dots, X_n,</math> denoted <math>K[X_1,\dots, X_n],</math> is thus a vector space (or a free module, if Template:Mvar is a ring) that has the monomials as a basis.
<math>K[X_1,\dots, X_n]</math> is naturally equipped (see below) with a multiplication that makes a ring, and an associative algebra over Template:Mvar, called the polynomial ring in Template:Mvar indeterminates over Template:Mvar (the definite article the reflects that it is uniquely defined up to the name and the order of the indeterminates. If the ring Template:Mvar is commutative, <math>K[X_1,\dots, X_n]</math> is also a commutative ring.
Operations in Template:MathEdit
Addition and scalar multiplication of polynomials are those of a vector space or free module equipped by a specific basis (here the basis of the monomials). Explicitly, let <math>p=\sum_{\alpha\in I}p_\alpha X^\alpha,\quad q=\sum_{\beta\in J}q_\beta X^\beta,</math> where Template:Mvar and Template:Mvar are finite sets of exponent vectors.
The scalar multiplication of Template:Mvar and a scalar <math>c\in K</math> is
- <math>cp = \sum_{\alpha\in I}cp_\alpha X^\alpha.</math>
The addition of Template:Mvar and Template:Mvar is
- <math>p+q = \sum_{\alpha\in I\cup J}(p_\alpha+q_\alpha) X^\alpha,</math>
where <math>p_\alpha=0</math> if <math>\alpha \not\in I,</math> and <math>q_\beta=0</math> if <math>\beta \not\in J.</math> Moreover, if one has <math>p_\alpha+q_\alpha=0</math> for some <math>\alpha \in I \cap J,</math> the corresponding zero term is removed from the result.
The multiplication is
- <math>pq = \sum_{\gamma\in I+J}\left(\sum_{\alpha, \beta\mid \alpha+\beta=\gamma} p_\alpha q_\beta\right) X^\gamma,</math>
where <math>I+J</math> is the set of the sums of one exponent vector in Template:Mvar and one other in Template:Mvar (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 an associative algebra is straightforward.
Polynomial expressionEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }} }} A polynomial expression is an expression built with scalars (elements of Template:Mvar), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.
As all these operations are defined in <math>K[X_1,\dots, X_n]</math> a polynomial expression represents a polynomial, that is an element of <math>K[X_1,\dots, X_n].</math> The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial. Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the distributive law all the products that have a sum among their factors, and then using commutativity (except for the product of two scalars), and associativity for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the like terms.
The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of computer algebra, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.
Categorical characterizationEdit
Template:Anchor If Template:Mvar is a commutative ring, the polynomial ring Template:Math has the following universal property: for every [[commutative algebra (structure)|commutative Template:Mvar-algebra]] Template:Mvar, and every Template:Mvar-tuple Template:Math of elements of Template:Mvar, there is a unique algebra homomorphism from Template:Math to Template:Mvar that maps each <math>X_i</math> to the corresponding <math>x_i.</math> This homomorphism is the evaluation homomorphism that consists in substituting <math>X_i</math> with <math>x_i</math> in every polynomial.
As it is the case for every universal property, this characterizes the pair <math>(K[X_1, \dots, X_n], (X_1, \dots, X_n))</math> up to a unique isomorphism.
This may also be interpreted in terms of adjoint functors. More precisely, let Template:Math and Template:Math be respectively the categories of sets and commutative Template:Mvar-algebras (here, and in the following, the morphisms are trivially defined). There is a forgetful functor <math>\mathrm F: \mathrm{ALG}\to \mathrm{SET}</math> that maps algebras to their underlying sets. On the other hand, the map <math>X\mapsto K[X]</math> defines a functor <math>\mathrm{POL}: \mathrm{SET}\to \mathrm{ALG}</math> in the other direction. (If Template:Mvar is infinite, Template:Math is the set of all polynomials in a finite number of elements of Template:Mvar.)
The universal property of the polynomial ring means that Template:Math and Template:Math are adjoint functors. That is, there is a bijection
- <math>\operatorname{Hom}_{\mathrm {SET}}(X,\operatorname{F}(A))\cong \operatorname{Hom}_{\mathrm {ALG}}(K[X], A). </math>
This may be expressed also by saying that polynomial rings are free commutative algebras, since they are free objects in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.
Graded structureEdit
Every polynomial ring is a graded ring: one can write the polynomial ring <math>R = K[X_1, \ldots, X_n]</math> as a direct sum <math display=block> R = \bigoplus_{i = 0}^\infty R_i</math> where <math>R_i</math> is the subspace consisting of all homogeneous polynomials of degree <math>i</math> (along with the zero polynomial); then for any elements <math>f \in R_i</math> and <math>g \in R_j</math>, their product <math>fg</math> belongs to <math>R_{i + j}</math>.
Univariate over a ring vs. multivariateEdit
A polynomial in <math>K[X_1, \ldots, X_n]</math> can be considered as a univariate polynomial in the indeterminate <math>X_n</math> over the ring <math>K[X_1, \ldots, X_{n-1}],</math> by regrouping the terms that contain the same power of <math>X_n,</math> that is, by using the identity
- <math>\sum_{(\alpha_1, \ldots, \alpha_n)\in I} c_{\alpha_1, \ldots, \alpha_n} X_1^{\alpha_1} \cdots X_n^{\alpha_n}=\sum_i\left(\sum_{(\alpha_1, \ldots, \alpha_{n-1})\mid (\alpha_1, \ldots, \alpha_{n-1}, i)\in I} c_{\alpha_1, \ldots, \alpha_{n-1}} X_1^{\alpha_1} \cdots X_{n-1}^{\alpha_{n-1}}\right)X_n^i,</math>
which results from the distributivity and associativity of ring operations.
This means that one has an algebra isomorphism
- <math>K[X_1, \ldots, X_n]\cong (K[X_1, \ldots, X_{n-1}])[X_n]</math>
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 a unique 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, by induction on the number of indeterminates.
The main such properties are listed below.
Properties that pass from Template:Math to Template:MathEdit
In this section, Template:Mvar is a commutative ring, Template:Mvar is a field, Template:Mvar denotes a single indeterminate, and, as usual, <math>\mathbb Z</math> is the ring of integers. Here is the list of the main ring properties that remain true when passing from Template:Mvar to Template:Math.
- If Template:Mvar is an integral domain then the same holds for Template:Math (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors).
- In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are integral domains.
- If Template:Mvar is a unique factorization domain then the same holds for Template:Math. This results from Gauss's lemma and the unique factorization property of <math>L[X],</math> where Template:Mvar is the field of fractions of Template:Mvar.
- In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are unique factorization domains.
- If Template:Mvar is a Noetherian ring, then the same holds for Template:Math.
- In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are Noetherian rings; this is Hilbert's basis theorem.
- If Template:Mvar is a Noetherian ring, then <math>\dim R[X] = 1+\dim R,</math> where "<math>\dim</math>" denotes the Krull dimension.
- In particular, <math>\dim K[X_1,\ldots,X_n] = n</math> and <math>\dim \mathbb Z[X_1,\ldots,X_n] = n+1.</math>
- If Template:Mvar is a regular ring, then the same holds for Template:Math; in this case, one has <math display="block">\operatorname{gl}\, \dim R[X]= \dim R[X]= 1 + \operatorname{gl}\, \dim R=1+\dim R,</math> where "<math>\operatorname{gl}\, \dim</math>" denotes the global dimension.
- In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are regular rings, <math>\operatorname{gl}\, \dim \mathbb Z[X_1,\ldots,X_n] = n+1,</math> and <math>\operatorname{gl}\, \dim K[X_1,\ldots,X_n] = n.</math> The latter equality is Hilbert's syzygy theorem.
Several indeterminates over a fieldEdit
Polynomial rings in several variables over a field are fundamental in invariant theory and algebraic 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 under affine or projective 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 and Jacobian conjecture are among the most famous properties that are specific to multivariate polynomials over a field.
Hilbert's NullstellensatzEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by David Hilbert, which extends to the multivariate case some aspects of the fundamental theorem of algebra. It is foundational for algebraic geometry, as establishing a strong link between the algebraic properties of <math>K[X_1, \ldots, X_n]</math> and the geometric properties of algebraic varieties, that are (roughly speaking) set of points defined by implicit 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 a complex zero if and only if it is not a constant. The statement is: a set of polynomials Template:Mvar in <math>K[X_1, \ldots, X_n]</math> has a common zero in an algebraically closed field containing Template:Mvar, if and only if Template:Math does not belong to the ideal generated by Template:Mvar, that is, if Template:Math is not a linear combination of elements of Template:Mvar with polynomial coefficients.
The second version generalizes the fact that the irreducible univariate polynomials over the complex numbers are associate to a polynomial of the form <math>X-\alpha.</math> The statement is: If Template:Mvar is algebraically closed, then the maximal ideals of <math>K[X_1, \ldots, X_n]</math> have the form <math>\langle X_1 - \alpha_1, \ldots, X_n - \alpha_n \rangle.</math>
Bézout's theoremEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Bézout's theorem may be viewed as a multivariate generalization of the version of the fundamental theorem of algebra that asserts that a univariate polynomial of degree Template:Mvar has Template:Mvar complex roots, if they are counted with their multiplicities.
In the case of bivariate polynomials, it states that two polynomials of degrees Template:Mvar and Template:Mvar in two variables, which have no common factors of positive degree, have exactly Template:Mvar common zeros in an algebraically closed field containing the coefficients, if the zeros are counted with their multiplicity and include the zeros at infinity.
For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with homogeneous polynomials, and consider zeros in a projective space. In this context, a projective zero of a homogeneous polynomial <math>P(X_0, \ldots, X_n)</math> is, up to a scaling, a Template:Math-tuple <math>(x_0, \ldots, x_n)</math> of elements of Template:Mvar that is different from Template:Math, and such that <math>P(x_0, \ldots, x_n) = 0 </math>. Here, "up to a scaling" means that <math>(x_0, \ldots, x_n)</math> and <math>(\lambda x_0, \ldots, \lambda x_n)</math> are considered as the same zero for any nonzero <math>\lambda\in K.</math> In other words, a zero is a set of homogeneous coordinates of a point in a projective space of dimension Template:Mvar.
Then, Bézout's theorem states: Given Template:Mvar homogeneous polynomials of degrees <math>d_1, \ldots, d_n</math> in Template:Math indeterminates, which have only a finite number of common projective zeros in an algebraically closed extension of Template:Mvar, the sum of the multiplicities of these zeros is the product <math>d_1 \cdots d_n.</math>
Jacobian conjectureEdit
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GeneralizationsEdit
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 polynomial rigs.
Infinitely many variablesEdit
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 the free commutative algebra, the only difference is that it is a free 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 the ring of power series in infinitely many variables. Such a ring is used for constructing the ring of symmetric functions over an infinite set.
Generalized exponentsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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: Template:Nowrap. A set for which addition makes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if Template:Nowrap, then Template:Nowrap for every n in N. The multiplication is defined as the Cauchy product, so that if Template:Nowrap, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n.
When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:
- <math>\sum_{n \in N} a_n X^n</math>
and then the formulas for addition and multiplication are the familiar:
- <math>\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</math>
and
- <math>\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</math>
where the latter sum is taken over all i, j in N that sum to n.
Some authors such as Template:Harv go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N 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 taking N to be the additive monoid of non-negative rational numbers, Template:Harv. See also Puiseux series.
Power seriesEdit
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Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N 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 of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the ring completion of the polynomial ring with respect to the ideal generated by Template:Mvar.
Noncommutative polynomial ringsEdit
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For polynomial rings of more than one variable, the products X⋅Y and Y⋅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 in n noncommuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of n 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 in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.
Differential and skew-polynomial ringsEdit
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Other generalizations of polynomials are differential and skew-polynomial rings.
A differential polynomial ring is a ring of differential operators formed from a ring R and a derivation δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation Template:Nowrap may be rewritten as
- <math>X\cdot a = a\cdot X +\delta(a).</math>
This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a noncommutative ring.
The standard example, called a Weyl algebra, takes R to be a (usual) polynomial ring k[Y ], and δ to be the standard polynomial derivative <math>\tfrac{\partial}{\partial Y}</math>. Taking a = Y in the above relation, one gets the canonical commutation relation, X⋅Y − Y⋅X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra. Template:Harv.
The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the endomorphism ring of R, the formula Xn⋅r = F(n)(r)⋅Xn allows constructing a skew-polynomial ring. Template:Harv Skew polynomial rings are closely related to crossed product algebras.
Polynomial rigsEdit
Template:See also The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig. For example, the set of all multivariate polynomials with natural number coefficients is a polynomial rig.