Fundamental Theorem of Symmetric Polynomials

1. Fundamental Theorem of Symmetric Polynomials Mu Zhao, David Li, Chao Lu Towson University Jon A. Sjogren Air Force Office of Scientific Research

2. Theorem: • The fundamental theorem of symmetric polynomials, it states that any symmetric polynomial in N variables can be given by a polynomial expression in terms of these elementary symmetric polynomials.

3. Proof: • Step 1: Given a symmetric polynomial, sort terms of the polynomial by lexicographic order. Example 1: Given a symmetric polynomial P: the lexicographic order in P is as following:

4. Proof: • Step 2: Find the largest monomial, then find product of elementary symmetric polynomials which could eliminate the largest monomial according to this order without introducing any larger monomials. In example 1: The largest monomial in P is . The product of elementary symmetric polynomials which can be used to eliminate is . where

5. Proof: • Step 3: Reduce P into elementary symmetric polynomials by successively subtracting from P a product of elementary symmetric polynomials eliminating the largest monomial according to this order without introducing any larger monomials. This way, in each step, the largest monomial becomes smaller and smaller until it becomes zero. (Cont’d)

6. Proof: Now, since becomes the largest monomial, repeat step2 to find the product of elementary symmetric polynomials can eliminate , which is . Then, So, P can be rewritten as END

7. Example 2: • If there is polynomial as , are roots of it. To expand the expression, we could also get: where DEMO • If there is another polynomial Q(x) as: Now, Q(x) is function of , it can be expressed as function of .

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9. Example 2: • Step 1: Let • Substitute into Q, so that DEMO

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11. Example 2: • Step 2: Expand Q as where DEMO

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16. Example 2: • Step 3: Now, is function of . As long as can be represented as a function of , which is certain because is symmetric polynomial of , can be rewritten as function of . DEMO

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18. Generalization • Our program can be used to solve similar problem with different degree or different T categories. • Change degree: • Change T category: DEMO

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20. Question: • In which field, this fundamental theorem of symmetric polynomial can be applied? We like your suggestions on applications.

21. Why we build our own library? • It is more flexible to tailor the functions to the user’s need. Example: Fundamental theorem of symmetric polynomials. • Existing software has limitations on solving the problems. Example: They are not exact computation, they have rounding error problem even in symbolic software.