WORKSHEET ON SYMMETRIC POLYNOMIALS DUE WEDNESDAY, APRIL 4 1. Elementary symmetric polynomials Definition 1.1. Let R be a ring (commutative, with unit). A polynomial f 2 R[x1; : : : ; xn] is symmetric if for any σ 2 Sn, f(σ(x1); : : : ; σ(xn)) = f(x1; : : : ; xn) Alternatively, define the action of Sn on R[x1; : : : ; xn] via σ ◦ f(x1; : : : ; xn) = f(σ(x1); : : : ; σ(xn)): The symmetric polynomials are invariants of this action - the polynomials for which the stabilizer is the entire group Sn. 17 17 17 16 16 16 Example 1.2. Let n = 3. Then x1 +x2 +x3 , x1x2 +x2x3 +x3x1 are symmetric 2 3 whereas x1x2x3 is not. Consider the polynomial P (t) = (t − x1)(t − x2) ::: (t − xn) in R[x1; : : : ; xn][t]. Let n n−1 n−2 n P (t) = t − s1(x1; : : : ; xn)t + s2(x1; : : : ; xn)t − ::: + (−1) sn(x1; : : : ; xn) Definition 1.3. Polynomials si(x1; : : : ; xn), 1 ≤ i ≤ n, are called the elementary symmetric polynomials. Observe that P (t) is clearly invariant under the action of Sn. Hence, the ele- mentary symmetric polynomials are, in fact, symmetric. Of course, one can write them down explicitly: s1 = x1 + ::: + xn s2 = x1x2 + x1x3 + ::: + xn−1xn ::: sn = x1 : : : xn n α α1 αn Let α = (α1; : : : ; αn) 2 Z≥0 and denote by x the monomial x1 : : : xn . We'll say that xα > xβ if α > β in lexicographical order. If f is a polynomial in R[x1; : : : ; xn] then the multidegree of f is the degree α of the maximal monomial α in f. The degree of a monomial x is α1 + ::: + αn. The degree of a polynomial f is the degree of its highest monomial. α P σ(α1) σ(αn) Observe that any symmetric polynomial containing x must contain x1 : : : xn . σ2Sn Definition 1.4. A polynomial f is called homogeneous if f is a sum of monomials of the same degree. Note that elementary symmetric polynomials are homogeneous and determined by a multidegree α which consists of only 0's and 1's. 1 2 DUE WEDNESDAY, APRIL 4 Theorem 1.5. (= Problem 1) Let f(x1; : : : ; xn) 2 R[x1; : : : ; xn] be a symmetric polynomial. Then there exists a polynomial F 2 R[x1; : : : ; xn] such that f(x1; : : : ; xn) = F (s1; : : : ; sn). In other words, any symmetric polynomial can be expressed in terms of elemen- tary ones. 3 3 3 3 Example 1.6. x1 + x2 + x3 = s1 − 3s1s2 + 3s3: Definition 1.7. We say that f1; : : : ; fm 2 R[x1; : : : ; xn] are algebraically indepen- dent if there does not exist F 2 R[x1; : : : ; xm] such that F (f1; : : : ; fm) = 0. Theorem 1.8. (=Problem 2). Prove that elementary symmetric polynomials on n variables are algebraically independent. The combination of these two results is sometimes referred to as the \Funda- mental Theorem of symmetric polynomials": Theorem 1.9. The ring of invariants of the polynomial ring on n variables under the action of the symmetric group is a polynomial ring on the elementary symmetric polynomials: Sn R[x1; : : : ; xn] ' R[s1; : : : ; sn]: Sn R[x1; : : : ; xn] is called the ring of symmetric polynomials. 2. Newton identities k k Let pk(x1; : : : ; xn) = x1 +:::+xn. By the previous theorem, pk can be expressed in terms of elementary symmetric polynomials. Explicit formulas can be obtained recursively from the Newton Identities: k X i−1 ksk = (−1) sk−ipi i=1 Problem 3. (1) Prove Newton identities. Assume s0 = 1. (2) Assume R is a field of characteristic 0. Show that fp1; : : : ; png are al- gebraically independent generators of the ring of symmetric polynomials Sn R[x1; : : : ; xn] . Corollary 2.1. Let t1; : : : ; tn be all roots (counted with multiplicity and, possibly, k k complex) of a polynomial of degree n with real coefficients. Then t1 + ::: + tn is a real number for any k. Problem 4. Let A be a real-valued matrix. Show that the characteristic polyno- mial χ(A) can be expressed exclusively in terms of Tr(Ak) for k ≥ 1..