Wikipedia:Reference desk/Archives/Mathematics/2013 November 30

Mathematics desk
< November 29 << Oct | November | Dec >> Current desk >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


November 30

edit

Adding polynomial closed form expressions

edit

Is there a way to sum together two polynomial equations of the form   and obtain a new equation, without solving for x? To clarify, the root(s) of the resulting polynomial should be the arithmetic sum(s) of the root(s) of the original polynomials and what I want to do is to sum two lists of coefficients   to obtain one list with the proper coefficients for the new polynomial whose roots have been summed from the originals. Ginsuloft (talk) 11:54, 30 November 2013 (UTC)[reply]

So for instance given two quintics   with roots   and   with roots   to give a polynomial with roots  ?
I don't believe that is possible as one can't give an order to the roots so we don't know which of each is 1 and which is 5. However it may be possible, I'd have to think a bit more about this, to do what you want with a polynomial result where every combination of sums was a root, i.e. it would include   for instance. For the quintics this would give a polynomial of order 25. Dmcq (talk) 13:31, 30 November 2013 (UTC)[reply]
Yes you can get an order 25 polynomial using sums and products the coefficients of the quintics where the roots are all the pairwise sums of the original quintics, or in general get a polynomial of order mn like that for any two polynomials of order m and n. Dmcq (talk) 14:05, 30 November 2013 (UTC)[reply]
To be more specific, write both polynomials as the characteristic polynomial of a matrix, this isn't hard to do. Then you can combine the two matrices to get a new one whose eigenvalues are the sums of the eigenvalues of the two given ones. The simplest way of explaining this will be to do the degree 2 case. Let
a1 a2
a3 a4
and
b1 b2
b3 b4
be the two given matrices. Then eigenvalues of
a1+b1 a1+b2 a2+b1 a2+b2
a1+b3 a1+b4 a2+b3 a2+b4
a3+b1 a3+b2 a4+b1 a4+b2
a3+b3 a3+b4 a4+b3 a4+b4
are the sums of the eigenvalues of the given matrices. In the example you give the two starting matrices are
 0  1  0  0  0
 0  0  1  0  0
 0  0  0  1  0
 0  0  0  0  1
-1 -1  0  0  0
and
 0  1  0  0  0
 0  0  1  0  0
 0  0  0  1  0
 0  0  0  0  1
-2 -1  0  0  0
Combine these as above, (my screen is too small to write out a 25x25 matrix) to get a matrix whose characteristic polynomial is the polynomial you want. This has some important implications, for example you use this idea to prove that the ring of algebraic integers is, in fact, a ring. --RDBury (talk) 15:08, 30 November 2013 (UTC)[reply]

There exist polynomials f and g such that the equations f(x)=g(y)=x+y−z=0 are satisfied by x and y and z=x+y. Eliminate x and y from these equations to obtain an equation h(z)=0 where h is a polynomial. Bo Jacoby (talk) 15:04, 3 December 2013 (UTC).[reply]

Summation sign

edit

Does Σ(xyz) mean x+y+z? 194.66.246.61 (talk) 20:54, 30 November 2013 (UTC)[reply]

No, xyz would still mean x×y×z inside a summation sign. Σ means a sum taken over something specified in the notation or sometimes implied by the context. For example,   "Σ(xyz)" by itself doesn't give enough context to guess what is being summed. PrimeHunter (talk) 21:04, 30 November 2013 (UTC)[reply]