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I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree <=5.

How do I express it in terms of the elementary symmetric functions s1=(c1+c2+c3+c4+c5), s2=(c1c2+c1c3+...+c4c5), s3=(c1c2c3+...+c3c4c5), s4=(c1c2c3c4+...+c2c3c4c5), s5=c1c2c3c4c5?

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    $\begingroup$ This looks confusing, and confused. For a function of the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5) to be symmetric requires p1, p2, p3, p4, p5 to be all identical. And then writing it in the elementaries is easy (because what you have is basically a decomposition in the power sums, and all it remains to do is decompose the power sums in the elementaries). The condition c1 < c2 < c3 < c4 < c5 seems out of place. What is the context of your question? $\endgroup$
    – darij grinberg
    Commented Oct 22, 2013 at 18:37
  • $\begingroup$ My function is only defined by that formula for c1, c2, c3, c4, c5 sorted. To calculate the function for arguments which are not sorted, first sort then and then apply the formula. By definition, this is a symmetric function. Its restriction to the 5-d region c1 < c2 < c3 < c4. < c5 is given by the formula I indicated. (And similarly if some of the < signs are replaced with <= signs). Since it IS symmetric, there must be an expression in terms of the elementary symmetric functions, which reduces to the formula when the inputs are sorted, but reduces to a different formula when the inputs have $\endgroup$
    – Joe Shipman
    Commented Oct 23, 2013 at 0:33
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    $\begingroup$ Joe, I converted your answer to a comment (since it is not an answer to your question). You can always comment under your own posts (you need a small amount of rep to comment on other people's posts). But, instead of (or in addition to) adding clarification through a comment, it is considered good practice to edit your own question so that the question is made clear. Also, we strongly encourage people to use LaTeX to format the mathematics. $\endgroup$
    – Todd Trimble
    Commented Oct 23, 2013 at 0:43
  • $\begingroup$ I would have except upon rereading I am satisfied that it was worded quite precisely and was simply misinterpreted because of a careless reading rather than a lack of clarity. $\endgroup$
    – Joe Shipman
    Commented Oct 23, 2013 at 3:13
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    $\begingroup$ ... elementaries. See, e. g., en.wikipedia.org/wiki/… . $\endgroup$
    – darij grinberg
    Commented Oct 23, 2013 at 5:07

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Since you don't specify the form you want the answer in, there's a cheap answer: Your function is given by "$p_1$ of the smallest root of $x^5-s_1x^4+s_2x^3-s_3x^2+s_4x-s_5$, plus $p_2$ of the second smallest root, plus $p_3$ of the middle root, plus $p_4$ of the second-largest root, plus $p_5$ of the largest root." I assume this isn't the sort of thing you had in mind, but I'm not sure what additional requirements you want to impose on an answer. You can't expect a really nice formula, because your function is unlikely to be smooth at places where two of the $c_i$'s are equal.

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  • $\begingroup$ How did you arrive at that? If I knew a bound on the degree of the answer (either total degree as a multivariate polynomial in the s_i, or weighted total degree corresponding to the degree of the associated polynomial in the c_i), I could just use undetermined coefficients and invert a big matrix, I was looking for an answer of that sort. $\endgroup$
    – Joe Shipman
    Commented Oct 23, 2013 at 3:06
  • $\begingroup$ @JoeShipman The answer won't generally be a polynomial in the $s_i$. Consider a much simpler case (2 instead of 5 variables but otherwise analogous to your question): Define $f(x,y)=y$ when $x<y$, and make it symmetric (and continuous). So $f(x,y)=\max\{x,y\}$. That's certainly not a polynomial in $s_1=x+y$ and $s_2=xy$ (because such a polynomial would also be a polynomial in $x$ and $y$, which $f$ isn't because its gradient is discontinuous on the line $x=y$). $\endgroup$
    – Andreas Blass
    Commented Oct 23, 2013 at 20:39
  • $\begingroup$ I already gave exactly that example in another comment. I asked for an expression in terms of the elementary symmetric polynomials s_i. I did not require that it be a polynomial in the s_i. My example was |x1-x2| and the expression I gave was sqrt(s1^2 - 4s2) which is obviously not a polynomial. The fact that the expression must be integral when the original variables are integral suggests it cannot be too horrible. $\endgroup$
    – Joe Shipman
    Commented Oct 24, 2013 at 1:03
  • $\begingroup$ @JoeShipman Now I'm really confused about what you want. You're aware that the answer needn't be a polynomial, yet you refer (in your first comment under my answer) to "the degree of the answer" and (in a parenthetical comment) say other things that don't look meaningful to me unless the answer is a polynomial. $\endgroup$
    – Andreas Blass
    Commented Oct 24, 2013 at 16:29
  • $\begingroup$ I want a limit on the complexity of the expression, as a root of a poly whose coefficients are symmetric polys: a bound on the degree of the polynomial and on the degree and total-degree of its coefficients. E.g., |x1-x2| is a symmetric function which can be expressed as sqrt(s1^2 - 4s2) for the elementary symmetric functions s1=x1+x2, s2=x1x2, which is a root of the polynomial y^2 - (s1^2 - 4s2) = 0, or alternatively of the polynomial y^2 - (x1^2 - 2x1x2 + x2^2) = 0. This expression involves a degree-2 polynomial whose coefficients are symmetric polynomials of degree <=2 and total-degree <=2. $\endgroup$
    – Joe Shipman
    Commented Oct 24, 2013 at 20:11

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