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

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November 30

Adding polynomial closed form expressions

Is there a way to sum together two polynomial equations of the form $a_{n}x^{n}\ldots a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0$ 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 $a_{n}\ldots a_{0}$ 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)

So for instance given two quintics $x^{5}+x+1$ with roots $r_{1}\dots r_{5}$ and $x^{5}+x+2$ with roots $s_{1}\dots s_{5}$ to give a polynomial with roots $r_{1}+s_{1}\dots r_{5}+s_{5}$?

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 $r_{2}+s_{4}$ for instance. For the quintics this would give a polynomial of order 25. Dmcq (talk) 13:31, 30 November 2013 (UTC)

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)

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)

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).

Summation sign

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

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, $\sum _{i=3}^{6}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}=86.$ "Σ(xyz)" by itself doesn't give enough context to guess what is being summed. PrimeHunter (talk) 21:04, 30 November 2013 (UTC)