§ Fundamental theorem of symmetric polynomials
- Every symmetric polynomial of variables can be written in terms of the elementary symmetric polynomials , , . Generalize appropriately.
§ Two variable case
- For even further simplicity, consider the two variable case: every symmetric polynomial of variables can be written in terms of the elementary symmetric polynomials , , .
- Consider some symmetric polynomial . Define an ordering on its monomials iff either , or , or . So we compare first by degree, then by lexicographically. Thus, call this order the lex order.
bigmon(p) to be the largest monomial in . Define
bigcoeff(p) to be the coefficient of
bigmon(p). Finally, define
bigterm(p) = bigmon(p) . bigcoeff(p) as the leading term of the polynomial .
- Prove an easy theorem that
bigterm(pq) = bigterm(p)bigterm(q).
- Now, suppose we have a leading monomial . Actually, this is incorrect! If we have a monomial , then we will also have a monomial , which is lex larger than . Thus, in any leading monomial, we will have the powers be in non-increasing (decreasing order).
- OK, we have leading monomial . We wish to write this in terms of elementary symmetric polynomials. We could try and write this by using the leading term in and leading term in .
- This means we need to solve , or and . This tells us that we should choose and . If we do this, then our combination of symmetric polynomials will kill the leading term of . Any new terms we introduce will be smaller than the leading term, which we can write as elementary symmetric polynomials by induction!
§ Three variable case
- Now consider three variables. Once again, suppose has leading monomial . We saw earlier that we must have for it to be a leading monomial.
- Let's write it in terms of . So we want to write it as product of , , and . Their product is .
- This gives us the system of equations . This means (1) , (2) or , and (3) or .
§ General situation
- think of the monomial as a vector . Then the leading terms of the symmetric polynomials correspond to , , and ].
- When we take powers of symmetric polynomials, we scale their exponent vector by that power. So for example, the leading term of is which is .
- When we multiply symmetric polynomials, we add their exponent vectors. For example, the leading term of is . This is equal to
- Thus, we wish to write the vector as a linear combination of vectors , , and . This is solving the equation:
[a] [1 1 1][p]
[b] = [0 1 1][q]
[c] [0 0 1][r]
- subject to the conditions that the unknowns , given knowns such that .
- Let's check that the equation makes sense: as all of , and have a at the position. Similarly for .
- Solve by the usual back-substitution.