Timeline for Looking for product of symmetric polynomials evaluated at roots of unity
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 20, 2015 at 18:26 | history | edited | Wolfgang | CC BY-SA 3.0 |
added 1 character in body
|
| Sep 25, 2013 at 12:45 | comment | added | Turbo | I issue is I do not know what Schur bassis are and how Schur basis is useful here. | |
| Sep 25, 2013 at 11:37 | comment | added | Per Alexandersson | Oh, I must have misinterpreted the question a bit, but ok, try to expand the product in $a_1,\cdots,a_n$, and then express in the basis of Schur polynomials in $a_1,\dotsc,a_n$. This might give some insight.... | |
| Sep 24, 2013 at 13:04 | comment | added | Turbo | @PerAlexandersson could you explain what you mean by schur basis and so on as an answer? | |
| Jul 7, 2013 at 10:00 | comment | added | Per Alexandersson | (Some) symmetric polynomials really "like" having roots of unity. I know this statement is rather vague, but I have done some extensive experiments on common roots of certain Schur polynomials, and it is VERY common to have roots of unity as common roots. I believe this is due to the Vandermonde formula, so my tip is to try to express your things in Schur basis, and then exploit the Vandermonde expression for Schur polynomials. | |
| Jul 7, 2013 at 8:27 | history | edited | Turbo |
edited tags
|
|
| Nov 6, 2010 at 5:34 | comment | added | Turbo | From this view point $\frac{1}{N}\displaystyle \sum_{r=0}^{N-1}\Pi(\alpha^{r})\alpha^{-r(N-i)} = $ the $i$th term of the product of all the polynomials. Hence: $\frac{1}{N}\displaystyle \sum_{r=0}^{N-1}\Pi(\alpha^{r}) = 1$ and $\frac{1}{N}\displaystyle \sum_{r=0}^{N-1}\Pi(\alpha^{r})\alpha^{-r(N-t)} = 0$ $\forall t \in [1,n]$ with $n \ge 1$. | |
| Nov 6, 2010 at 4:34 | comment | added | Turbo | one more thing that should be known about the equations in that each term $(a_{i_{1}} + a_{i_{2}}+\cdots+\a_{i_{t}})$ correspond to FFT of size $N$ of some polynomial with coefficients $0/1$ with maximum degree $n$ -$\displaystyle \sum_{j=1}^{t}x^{i_{j}}$ (the top $N-n$ elements of the vector to be FFTed will be zero). Taking all such products would correspond to FFT of product of all degree $\le$ $n$ polynomials. This is the product $\Pi(\alpha)$. Taking inverse FFT of the products would give the product of all polynomials with degree $\le n$. This is kind of polynomial version of factorial. | |
| Nov 5, 2010 at 6:19 | comment | added | Turbo | No I have pretty much told what I know. | |
| Nov 5, 2010 at 4:41 | comment | added | Gerry Myerson | Thanks for the clarifications. It sounds like you have done some computations. Why not let people know what you have found, to save us the effort of repeating what you've already done? | |
| Nov 5, 2010 at 2:02 | comment | added | Turbo | "....product of all degree $\le n$ polynomials with $0/1$..." | |
| Nov 5, 2010 at 2:01 | history | edited | Turbo |
edited tags
|
|
| Nov 5, 2010 at 1:45 | comment | added | Turbo | One more thing I am almost sure is $\frac{1}{N}\displaystyle \sum_{r=0}^{N-1}\Pi(\alpha^{r}) = 1$, $\frac{1}{N}\displaystyle \sum_{r=0}^{N-1}\Pi(\alpha^{r})\alpha^{r(N-t)} = 0$ $\forall t \in [1,n]$ with $n \ge 1$. | |
| Nov 5, 2010 at 1:41 | history | edited | Turbo | CC BY-SA 2.5 |
edited body
|
| Nov 5, 2010 at 1:26 | history | edited | Turbo | CC BY-SA 2.5 |
added 62 characters in body; deleted 66 characters in body
|
| Nov 5, 2010 at 1:01 | comment | added | Turbo | Also product of any two of the Monomial symmetric polynomial seem to have some relation. See 1.4 of this: mathcircle.berkeley.edu/BMC3/SymPol.pdf | |
| Nov 5, 2010 at 0:55 | comment | added | Turbo | $N = 2 + (n-1)2^{n+1}$ comes from degree of product of all degree $n$ polynomials with $0$/$1$ coefficients. I think the product of $P_{i}$ can be thought of product of Monomial_symmetric_polynomials. en.wikipedia.org/wiki/… My belief that there may be relation bw diff prod comes from the fact that the symm poly are related to power symm poly. Also angle of prod seem to have simple relation. I felt the magnitude might also have nice relation. I am only an engineer. There might be connections to alg geom & num theory that I dont see. | |
| Nov 5, 2010 at 0:38 | comment | added | Gerry Myerson | Do you have some reason to think there is an elementary expression? Does it come out nicely for small values of $n$? Is there any particular reason to be interested in this problem only in the case where $N=2+(n-1)2^{n+1}$? Is there any particular reason to be interested in this problem at all? Have you done any work on it? Do you have any partial results? Shouldn't it be tagged number theory, rather than rings-and-algebras? | |
| Nov 4, 2010 at 23:32 | history | edited | Turbo | CC BY-SA 2.5 |
added 23 characters in body
|
| Nov 4, 2010 at 23:17 | history | asked | Turbo | CC BY-SA 2.5 |