applying elementary symmetric polynomials
The method used in the proof of fundamental theorem of symmetric polynomials may be applied to concrete instances as follows.
We have four
for which the corresponding -products of the sum (1) are
respectively. Apparently, the first one is out of the question. Therefore, clearly
Using and makes , and , when
implying . Using similarly we get , , which give
yielding . Hence we have the result
Example 2. Let . If we suppose that , the possible highest terms are
whence we may write
For determining the coefficients, evidently we can put and in as follows.
. , , . Then we have , , , . Thus (3) gives .
. , . Now , , , , , whence (3) reads , giving .
. , . We get , , , . These yield , i.e. .
. , , . In this case, , , , , , whence , or . Consequently, we obtain from (3) the result
Although it has been derived by supposing (= the degree of ), it holds without this supposition. One has only to see that e.g. in the case , one must substitute to (4) the values , which changes the to the form
|Title||applying elementary symmetric polynomials|
|Date of creation||2013-03-22 19:10:07|
|Last modified on||2013-03-22 19:10:07|
|Last modified by||pahio (2872)|