HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a symmetric polynomial is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s are the most fundamental symmetric polynomials. A
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every ''symmetric'' polynomial expression in the roots of a
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\cd ...
can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and
Schur polynomial In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In ...
s play important roles alongside the elementary ones. The resulting structures, and in particular the
ring of symmetric functions In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which ...
, are of great importance in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and in
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
.


Examples

The following polynomials in two variables ''X''1 and ''X''2 are symmetric: :X_1^3+ X_2^3-7 :4 X_1^2X_2^2 +X_1^3X_2 + X_1X_2^3 +(X_1+X_2)^4 as is the following polynomial in three variables ''X''1, ''X''2, ''X''3: :X_1 X_2 X_3 - 2 X_1 X_2 - 2 X_1 X_3 - 2 X_2 X_3 There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is :\prod_(X_i-X_j)^2 where first a polynomial is constructed that changes sign under every exchange of variables, and taking the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant). On the other hand, the polynomial in two variables :X_1 - X_2 is not symmetric, since if one exchanges X_1 and X_2 one gets a different polynomial, X_2 - X_1. Similarly in three variables :X_1^4X_2^2X_3 + X_1X_2^4X_3^2 + X_1^2X_2X_3^4 has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric: :X_1^4X_2^2X_3 + X_1X_2^4X_3^2 + X_1^2X_2X_3^4 + X_1^4X_2X_3^2 + X_1X_2^2X_3^4 + X_1^2X_2^4X_3


Applications


Galois theory

One context in which symmetric polynomial functions occur is in the study of monic
univariate In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate ...
polynomials of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
''n'' having ''n'' roots in a given
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. These ''n'' roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the
fundamental theorem of symmetric polynomials In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
implies that a polynomial function ''f'' of the ''n'' roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
''f'' is given by a symmetric polynomial. This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s in the roots), how can one recover the roots? This leads to studying solutions of polynomials using the
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
of the roots, originally in the form of
Lagrange resolvents In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rati ...
, later developed in
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.


Relation with the roots of a monic univariate polynomial

Consider a monic polynomial in ''t'' of degree ''n'' :P=t^n+a_t^+\cdots+a_2t^2+a_1t+a_0 with coefficients ''a''''i'' in some field ''K''. There exist ''n'' roots ''x''1,…,''x''''n'' of ''P'' in some possibly larger field (for instance if ''K'' is the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, the roots will exist in the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s); some of the roots might be equal, but the fact that one has ''all'' roots is expressed by the relation :P = t^n+a_t^+\cdots+a_2t^2+a_1t+a_0=(t-x_1)(t-x_2)\cdots(t-x_n). By comparing coefficients one finds that :\begin a_&=-x_1-x_2-\cdots-x_n\\ a_&=x_1x_2+x_1x_3+\cdots+x_2x_3+\cdots+x_x_n = \textstyle\sum_x_ix_j\\ & \ \, \vdots\\ a_1&=(-1)^(x_2x_3\cdots x_n+x_1x_3x_4\cdots x_n+\cdots+x_1x_2\cdots x_x_n+x_1x_2\cdots x_) = \textstyle(-1)^\sum_^n\prod_x_j\\ a_0&=(-1)^nx_1x_2\cdots x_n. \end These are in fact just instances of
Viète's formulas In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). Basic formulas ...
. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression: although for a given polynomial ''P'' there may be qualitative differences between the roots (like lying in the base field ''K'' or not, being
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
or multiple roots), none of this affects the way the roots occur in these expressions. Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing ''P'', and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ''a''''i'' then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign (-1)^, are known as the
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s in ''x''1, …, ''x''''n''. A basic fact, known as the
fundamental theorem of symmetric polynomials In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
, states that ''any'' symmetric polynomial in ''n'' variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the ''coefficients'' of the polynomial, and in particular that its value lies in the base field ''K'' that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than ''K'' in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.


Special kinds of symmetric polynomials

There are a few types of symmetric polynomials in the variables ''X''1, ''X''2, …, ''X''''n'' that are fundamental.


Elementary symmetric polynomials

For each nonnegative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''k'', the elementary symmetric polynomial ''e''''k''(''X''1, …, ''X''''n'') is the sum of all distinct products of ''k'' distinct variables. (Some authors denote it by σ''k'' instead.) For ''k'' = 0 there is only the
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
so ''e''0(''X''1, …, ''X''''n'') = 1, while for ''k'' > ''n'', no products at all can be formed, so ''e''''k''(''X''1, ''X''2, …, ''X''''n'') = 0 in these cases. The remaining ''n'' elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts: *any symmetric polynomial ''P'' in ''X''1, …, ''X''''n'' can be written as a polynomial expression in the polynomials ''e''''k''(''X''1, …, ''X''''n'') with 1 ≤ ''k'' ≤ ''n''; *this expression is unique up to equivalence of polynomial expressions; *if ''P'' has
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
coefficients, then the polynomial expression also has integral coefficients. For example, for ''n'' = 2, the relevant elementary symmetric polynomials are ''e''1(''X''1, ''X''2) = ''X''1 + ''X''2, and ''e''2(''X''1, ''X''2) = ''X''1''X''2. The first polynomial in the list of examples above can then be written as :X_1^3+X_2^3-7=e_1(X_1,X_2)^3-3e_2(X_1,X_2)e_1(X_1,X_2)-7 (for a proof that this is always possible see the
fundamental theorem of symmetric polynomials In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
).


Monomial symmetric polynomials

Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic ''additive'' building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
, with only those copies required to obtain symmetry. Any monomial in ''X''1, …, ''X''''n'' can be written as ''X''1α1…''X''''n''α''n'' where the exponents α''i'' are
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s (possibly zero); writing α = (α1,…,α''n'') this can be abbreviated to ''X'' α. The monomial symmetric polynomial ''m''α(''X''1, …, ''X''''n'') is defined as the sum of all monomials ''x''β where β ranges over all ''distinct'' permutations of (α1,…,α''n''). For instance one has :m_(X_1,X_2,X_3)=X_1^3X_2X_3+X_1X_2^3X_3+X_1X_2X_3^3, :m_(X_1,X_2,X_3)=X_1^3X_2^2X_3+X_1^3X_2X_3^2+X_1^2X_2^3X_3+X_1^2X_2X_3^3+X_1X_2^3X_3^2+X_1X_2^2X_3^3. Clearly ''m''α = ''m''β when β is a permutation of α, so one usually considers only those ''m''α for which α1 ≥ α2 ≥ … ≥ α''n'', in other words for which α is a
partition of an integer In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...
. These monomial symmetric polynomials form a vector space
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
: every symmetric polynomial ''P'' can be written as a linear combination of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in ''P''. In particular if ''P'' has integer coefficients, then so will the linear combination. The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ ''k'' ≤ ''n'' one has :e_k(X_1,\ldots,X_n)=m_\alpha(X_1,\ldots,X_n) where α is the partition of ''k'' into ''k'' parts 1 (followed by ''n'' − ''k'' zeros).


Power-sum symmetric polynomials

For each integer ''k'' ≥ 1, the monomial symmetric polynomial ''m''(''k'',0,…,0)(''X''1, …, ''X''''n'') is of special interest. It is the power sum symmetric polynomial, defined as :p_k(X_1,\ldots,X_n) = X_1^k + X_2^k + \cdots + X_n^k . All symmetric polynomials can be obtained from the first ''n'' power sum symmetric polynomials by additions and multiplications, possibly involving
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficients. More precisely, :Any symmetric polynomial in ''X''1, …, ''X''''n'' can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials ''p''1(''X''1, …, ''X''''n''), …, ''p''''n''(''X''1, …, ''X''''n''). In particular, the remaining power sum polynomials ''p''''k''(''X''1, …, ''X''''n'') for ''k'' > ''n'' can be so expressed in the first ''n'' power sum polynomials; for example :p_3(X_1,X_2)=\textstyle\frac32p_2(X_1,X_2)p_1(X_1,X_2)-\frac12p_1(X_1,X_2)^3. In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in ''n'' variables with ''integral'' coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for ''n'' = 2, the symmetric polynomial :m_(X_1,X_2) = X_1^2 X_2 + X_1 X_2^2 has the expression : m_(X_1,X_2)= \textstyle\frac12p_1(X_1,X_2)^3-\frac12p_2(X_1,X_2)p_1(X_1,X_2). Using three variables one gets a different expression :\beginm_(X_1,X_2,X_3) &= X_1^2 X_2 + X_1 X_2^2 + X_1^2 X_3 + X_1 X_3^2 + X_2^2 X_3 + X_2 X_3^2\\ &= p_1(X_1,X_2,X_3)p_2(X_1,X_2,X_3)-p_3(X_1,X_2,X_3). \end The corresponding expression was valid for two variables as well (it suffices to set ''X''3 to zero), but since it involves ''p''3, it could not be used to illustrate the statement for ''n'' = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first ''n'' power sum polynomials involves rational coefficients may depend on ''n''. But rational coefficients are ''always'' needed to express elementary symmetric polynomials (except the constant ones, and ''e''1 which coincides with the first power sum) in terms of power sum polynomials. The
Newton identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomi ...
provide an explicit method to do this; it involves division by integers up to ''n'', which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of finite characteristic; however, it is valid with coefficients in any
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
containing the rational numbers.


Complete homogeneous symmetric polynomials

For each nonnegative integer ''k'', the complete homogeneous symmetric polynomial ''h''''k''(''X''1, …, ''X''''n'') is the sum of all distinct
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
s of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
''k'' in the variables ''X''1, …, ''X''''n''. For instance :h_3(X_1,X_2,X_3) = X_1^3+X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_2X_3+X_1X_3^2+X_2^3+X_2^2X_3+X_2X_3^2+X_3^3. The polynomial ''h''''k''(''X''1, …, ''X''''n'') is also the sum of all distinct monomial symmetric polynomials of degree ''k'' in ''X''1, …, ''X''''n'', for instance for the given example :\begin h_3(X_1,X_2,X_3)&=m_(X_1,X_2,X_3)+m_(X_1,X_2,X_3)+m_(X_1,X_2,X_3)\\ &=(X_1^3+X_2^3+X_3^3)+(X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_3^2+X_2^2X_3+X_2X_3^2)+(X_1X_2X_3).\\ \end All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in ''X''1, …, ''X''''n'' can be obtained from the complete homogeneous symmetric polynomials ''h''1(''X''1, …, ''X''''n''), …, ''h''''n''(''X''1, …, ''X''''n'') via multiplications and additions. More precisely: :Any symmetric polynomial ''P'' in ''X''1, …, ''X''''n'' can be written as a polynomial expression in the polynomials ''h''''k''(''X''1, …, ''X''''n'') with 1 ≤ ''k'' ≤ ''n''. :If ''P'' has integral coefficients, then the polynomial expression also has integral coefficients. For example, for ''n'' = 2, the relevant complete homogeneous symmetric polynomials are and . The first polynomial in the list of examples above can then be written as :X_1^3+ X_2^3-7 = -2h_1(X_1,X_2)^3+3h_1(X_1,X_2)h_2(X_1,X_2)-7. As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond ''h''''n''(''X''1, …, ''X''''n''), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased. An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities :\sum_^k(-1)^i e_i(X_1,\ldots,X_n)h_(X_1,\ldots,X_n) = 0, for all ''k'' > 0, and any number of variables ''n''. Since ''e''0(''X''1, …, ''X''''n'') and ''h''0(''X''1, …, ''X''''n'') are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the ''h''''k''(''X''1, …, ''X''''n'') with 1 ≤ ''k'' ≤ ''n'': one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.


Schur polynomials

Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.


Symmetric polynomials in algebra

Symmetric polynomials are important to
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
,
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, and
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
. They are also important in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, where they are mostly studied through the
ring of symmetric functions In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which ...
, which avoids having to carry around a fixed number of variables all the time.


Alternating polynomials

Analogous to symmetric polynomials are alternating polynomials: polynomials that, rather than being ''invariant'' under permutation of the entries, change according to the sign of the permutation. These are all products of the
Vandermonde polynomial In algebra, the Vandermonde polynomial of an ordered set of ''n'' variables X_1,\dots, X_n, named after Alexandre-Théophile Vandermonde, is the polynomial: :V_n = \prod_ (X_j-X_i). (Some sources use the opposite order (X_i-X_j), which changes the s ...
and a symmetric polynomial, and form a
quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.


See also

*
Symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\l ...
*
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
*
Stanley symmetric function In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric polynomials, symmetric functions introduced by in his study of the symmetric group of permutations. Formally, the Stanley symmetr ...
*
Muirhead's inequality In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. Preliminary definitions ''a''-mean For any real vector :a=(a_1,\dots ...


References

* * Macdonald, I.G. (1979), ''Symmetric Functions and Hall Polynomials''. Oxford Mathematical Monographs. Oxford: Clarendon Press. * I.G. Macdonald (1995), ''Symmetric Functions and Hall Polynomials'', second ed. Oxford: Clarendon Press. (paperback, 1998). *
Richard P. Stanley Richard Peter Stanley (born June 23, 1944) is an Emeritus Professor of Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. From 2000 to 2010, he was the Norman Levinson Professor of Applied Mathematics. He r ...
(1999), ''Enumerative Combinatorics'', Vol. 2. Cambridge: Cambridge University Press. {{Authority control Polynomials *