THE TEACHING OF MATHEMATICS 2012, Vol. XV, 1, pp. 55 59 THE FUNDAMENTAL THEOREM ON SYMMETRIC POLYNOMIALS Hamza Elhadi S. Daoub Abstract. In this work we are going to extend the proof of Newton s theorem of symmetric polynomials, by considering any monomial order > on polynomials in n variables x 1, x 2,, x n over a field k, where the original proof is based on the graded lexicographic order. We will introduce some basic definitions and propositions to support the extended proof. ZDM Subject Classification: H25; AMS Subject Classification: 97H20. Key words and phrases: Symmetric polynomials. The fundamental theorem of symmetric polynomials is, as its name suggests, a key result in the subject of root identities. The theorem states that any symmetric polynomial can be represented in terms of the elementary symmetric polynomials. Newton had made an extensive study of symmetric root polynomials as early as the mid-1660s, and was very likely aware of the fundamental theorem at that time. Edwards adds It must be admitted, however, that neither a careful statement nor a proof of it seems to have been published before the nineteenth century. Everyone seemed familiar with it and used it without inhibition. Kline credits Vandermonde with the first published proof of the fundamental theorem in 1771. However, it should be noted that Vandermonde s version of the result was stated in terms of roots and coefficients of a polynomial. Edwards makes the point that the fundamental theorem is properly about symmetric polynomials in n variables, independent of the context of coefficients and roots of a polynomial. Such a formulation sidesteps any philosophical issues concerning the existence or nature of roots. Definition 1.1. A polynomial f k[x 1, x 2,, x n ] is symmetric if f(x, x,, x ) = f(x 1, x 2,, x n ) for every possible permutation x, x,, x of the variables x 1, x 2,, x n. Definition 1.2. Given variables x 1, x 2,, x n, we define σ 1, σ 2,, σ n
56 H. E. S. Daoub k[x 1, x 2,, x n ] by σ 1 = x 1 + x 2 + + x n σ r = << <τ(r) σ n = x 1 x 2 x n x x x τ(r) And σ i is a symmetric polynomial for all i = 1,, n. Definition 1.3. A monomial ordering on k[x 1, x 2,, x n ] is any relation > on Z n 0, or equivalently, any relation on the set of monomials xα, α Z n 0 satisfying: i. > is a total ordering on Z n 0. ii. If α > β and γ Z 0 n, then α + γ > β + γ. iii. > is well-ordering on Z n 0. This means that every non-empty subset of Zn 0 has a smallest element under >. Proposition 1.4. An order relation > on Z 0 n if every strictly decreasing sequence in Z 0 n is a well-ordering if and only α(1) > α(2) > α(3) > eventually terminates. Definition 1.5. Let f = α a αx α be a nonzero polynomial in k[x 1, x 2,, x n ] and let > be a monomial order. The multidegree of f is multideg(f) = max(α Z n 0 : a α 0). The leading coefficient of f is LC(f) = a multideg(f) k. The leading monomial of f is LM(f) = x multideg(f). The leading term of f is LT (f) = LC(f) LM(f). Proposition 1.6. For f(x 1, x 2,, x n ), g(x 1, x 2,, x n ) k[x 1, x 2,, x n ] we have LT (fg) = LT (f)lt (g). Let > be any monomial order of the variables x 1, x 2,, x n such that x > x > > x. According to the Definition 1.3, we notice that x x > x x τ(i), where i = 3,, n x x x τ(3) > x x x τ(i), where i = 4,, n
The fundamental theorem on symmetric polynomials 57 and so on. The leading term of σ i is defined as follows: LT (σ 1 ) = x LT (σ 2 ) = x x LT (σ n ) = x x x Now, suppose that f k[x 1, x 2,, x n ] be any nonzero symmetric polynomial, and c 1 x a1 1 xa2 2 xan n be a leading term of f. Consider g = σ α1 1 σα2 2 σαn n. According to Proposition 1.6, the leading term of g is defined as follows: LT (g) = LT (σ α 1 1 σα 2 2 σαn n ) = LT (σ α1 1 )LT (σα2 2 ) LT (σα n n ) = (x ) α 1 (x x ) α 2 (x x x ) α n = x α 1+α 2 + +α n x α 2+ +α n x α n Therefore, we can find the values of α i, where LT (f) = LT (cg) c 1 x a1 1 xa2 2 xa n n a = α 1 + α 2 + + α n and a = α 2 + α 3 + + α n = c 1 x α1+α2+ +αn x α2+...αn x αn Thus, and a = α n Then, the map α n = a α n 1 = a τ(n 1) a α n 2 = a τ(n 2) a τ(n 1) α 2 = a a τ(3) α 1 = a a (a 1, a 2,, a n ) (a a, a a τ(3),, a τ(n 1) a, a ) defines the relation between the leading terms of f and g. Theorem 1.7. [Newton s Theorem] Any symmetric polynomial in k[x 1, x 2,, x n ] can be written as a polynomial in σ 1, σ 2,, σ n with coefficients in k and this polynomial is unique. Proof. We will follow the argument above. So let > be any monomial ordering of the variables x 1, x 2,, x n such that. x > x > > x
58 H. E. S. Daoub Let c 1 x a1 1 xa2 2 xa n n be a leading term of f, and consider g = σ a a 1 σ a a τ(3) 2 σ a n. LT (c 1 g) = LT (c 1 σ a a 1 σ a a τ(3) 2 σ a n ) = c 1 (x ) a a (x x ) a a τ(3) (x x x ) a = c 1 x a xa xa = LT (f) for any permutation τ. This shows that f and c 1 g have the same leading term. Hence f 1 = f c 1 g has a strictly smaller leading term according to the monomial ordering, which is defined above. Since f and g are symmetric polynomials, then so is f 1. Now we repeat this process, starting with f 1 instead of f, where f 1 has a leading term with coefficient c 2 and exponent b 1 < b 2 < < b n. As above, there is a symmetric polynomial g 1 such that f 1 and c 2 g 1 have the same leading term. It follows that f 2 = f 1 c 2 g 1 = f c 1 g c 2 g 1 has a strictly smaller leading term. Continuing in this way, we get polynomials: f, f 1 = f c 1 g, f 2 = f c 1 g c 2 g 1, f 3 = f c 1 g c 2 g 1 c 3 g 1, where at each stage the leading term gets smaller according to the ordering monomial. This process will terminate when we find some m with f m = 0; if not, then the above would give an infinite sequence of nonzero polynomials with strictly decreasing leading term, which contradicts Proposition 1.4. Hence, this process must terminate. Once we have f m = 0 for some m, we obtain f = c 1 g + c 2 g 1 + + c m g m 1 where each g i is a product of the σ i to various powers, which proves that f is a polynomial in the elementary symmetric polynomials. To prove the uniqueness, suppose that k[y 1, y 2,, y n ] be a polynomial ring with new variables y 1, y 2,, y n. Since the evaluation map which sends y i to σ i k[x 1, x 2,, x n ] defines a ring homomorphism ϕ : k[y 1, y 2,, y n ] k[x 1, x 2,, x n ] Then, if g = g(y 1, y 2,, y n ) k[y 1, y 2,, y n ], we get ϕ(g) = g(σ 1, σ 2,, σ n ) Recall that the set of all polynomials in the σ i with coefficient in k is a subring of k[x 1, x 2,, x n ], so we can write ϕ as a map ϕ : k[y 1, y 2,, y n ] k[σ 1, σ 2,, σ n ] where k[σ 1, σ 2,, σ n ] is the set of all polynomials in σ i with coefficients in k.
The fundamental theorem on symmetric polynomials 59 The last map is onto by the definition. To prove the uniqueness we need to prove that ϕ is one to one. It is sufficient to show that kerϕ = 0, which means for any nonzero polynomial g in y i, then g(σ 1, σ 2,, σ n ) 0. Let cu b 1 1 ub 2 2 ubn σ b 1 1 σb 2 2 σbn σ α 1 1 σα 2 2 σαn n n be any nonzero term in g. Applying ϕ gives g = n. As we mentioned before the theorem, the leading term of g = is cx b 1+b 2 + +b n x b 2+ +b n x b n. Since g is the sum of its terms, so the corresponding polynomial ϕ(g) is sum of cσ b 1 1 σb 2 2 σb n n. The crucial fact is that the map (b, b,, b ) (b 1 + b 2 + + b n, b 2 + + b n,, b n ) is one to one, so the leading terms cannot all cancel, and ϕ(g) cannot be the zeropolynomial. Hence the uniqueness follows. REFERENCES [1] D. Cox, J. Little, D. O Shea, Ideals, Varieties, and Algorithms, Springer Verlag UTM, New York, 1992. [2] D. A.Cox, Galois Theory, John Wiley and Sons, INC., New Jersey, 2004. [3] D. Kalman, Uncommon Mathematical Excursions, Polynomia and Related Realms, The Mathematical Association of America, Number 35, 2009. Faculty of Mathematics, University of Belgrade, Serbia