In mathematics, specifically in commutative algebra, elementary symmetric polynomials are the basic building blocks for symmetric polynomials, in the sense that every symmetric polynomial can be expressed as a sum of products of the elementary symmetric polynomials. Euclid, detail from The School of Athens by Raphael. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ...

Definition

The elementary symmetric polynomials in n variables can be defined as

and so forth, down to

For each positive integer, at most n, there exists exactly one elementary symmetric polynomial of degree k in n variables. To form the one which has degree k, we form all products of k-tuples of the n variables and add up these terms. The fact that x₁x₂ = x₂x₁ and so forth is the defining feature of commutative algebra. That is, the polynomial ring formed by taking all linear combinations of these variables is a commutative ring. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...

Properties

The elementary symmetric polynomials appear when we expand

That is, when we substitute values for the variables , we obtain the univariate polynomial whose roots are those values by plugging them into the elementary symmetric polynomials. Univariate describes a concept in statistics or econometrics. ...

In the case of the characteristic polynomial of a linear operator, the roots are the eigenvalues of the operator. When we plug these eigenvalues into the elementary symmetric polynomials, we obtain certain numerical invariants of the operator (namely, the coefficients of the characteristic polynomial). This is useful in tensor algebra and in disciplines which extensively employ tensor fields, such as differential geometry. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

The set of elementary symmetric polynomials in n variables generate the ring of symmetric polynomials in n variables. This is one of the foundations of invariant theory. In Abstract Algebra, a generator is defined as follows: Let G be a group and , then a is called a generator and G is a cyclic group. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ...

See also

• Newton's identities
• Schur polynomial
• Representation theory