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In my research project, I'm working with the following polynomial $$p(x) = 1 + 2x + 3x^2 + \ldots + nx^{n-1} + nx^n + nx^{n+1} + (n-1)x^{n+2} + (n-2)x^{n+3} + \ldots + 2x^{2n-1} + x^{2n}$$ or $$nx^n +\sum_{i=0}^{n-1}(i+1)x^{2n-i}+(i+1)x^i.$$ Now I would like this to be recognized as a polynomial by Mathematica, so I tried expanding it out with Expand[Sum[(i + 1) x^(2 n - i) + (i + 1) x^i, {i, 0, n - 1}] + n x^n]. That, however, computes the actual sum of the series, and it is no longer a polynomial.

So how do I make Mathematica just treat the sum "as is" so it is recognized as a polynomial? (Ultimately, I'm trying to determine if $p$ has roots of multiplicity $> 1$.)

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    $\begingroup$ I am not sure what you mean. What output do you expect to get? Does Total@Table[(i + 1) x^(2 n - i) + (i + 1) x^i, {i, 0, n - 1}] + n x^n do what you expect for any integer n>0? $\endgroup$
    – Somos
    Jul 10, 2023 at 23:32
  • $\begingroup$ Sorry for not being more clear — I use Mathematica very infrequently. I just want the polynomial above to "stay" a polynomial. I do not want Mathematica to evaluate it into a closed form result because after that I wish to take the resultant of $p$ and its reciprocal polynomial $p'$. The command you suggested returns an iterator does not have appropriate bounds error. $\endgroup$
    – wol
    Jul 10, 2023 at 23:40
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    $\begingroup$ Again, exactly what output do you expect to get? For any specific integer n>0 the code I gave produces a polynomial expression in x. Does using Defer[] help? As in Defer@Sum[(i + 1) x^(2 n - i) + (i + 1) x^i, {i, 0, n - 1}] + n x^n? $\endgroup$
    – Somos
    Jul 10, 2023 at 23:42
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    $\begingroup$ Well, ... is not a Mathematica expression. If that is what you want, then no Computer Algebra System can help you with this. $\endgroup$
    – Somos
    Jul 10, 2023 at 23:56
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    $\begingroup$ It seems that when n is even, -1 is a repeated root, and otherwise there are no repeated roots. $\endgroup$
    – Carl Woll
    Jul 11, 2023 at 14:07

2 Answers 2

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Your question is about a polynomial

p[n_] := n*x^n + Sum[ (i+1)*x^(2*n-i) + (i+1)*x^i, {i, 0, n-1}];
Factor[p[n]]
(* ((-1 + x^n)*(-1 + x^(2 + n)))/(-1 + x)^2 *)

which Mathematica is able to partially factor.

For any integer $\,n>0\,$ the polynomial $\,x^n-1\,$ has $\,n\,$ roots being all the $n$-th roots of unity. Thus it factors into $\,n\,$ linear factors, one for each $n$-th root of unity.

If $\,n>0\,$ is an integer then $\,x-1\,$ is a factor of $\,x^n-1\,$ because $1$ is a $n$-th root of unity. If $\,n>0\,$ is an even integer then $\,x+1\,$ is also a factor because $-1$ is also a $n$-th root of unity. In both cases, all other factors are complex conjugate pairs.

Now we define the polynomial $$ p_n := \frac{(x^n-1)(x^{n+2}-1)}{(x-1)^2}. $$ If $\,n>0 \,$ is even, then $\,n+2\,$ is also even which implies that $\,(x+1)^2\,$ is a factor of $\,p_n.\,$ If $\,n>0\,$ is odd, then $\,p_n\,$ has no real factors, only complex conjugate factors.

Your ultimate question

(Ultimately, I'm trying to determine if p has roots of multiplicity >1.)

has the answer yes if and only if $n$ is even.

Your penultimate question

So how do I make Mathematica just treat the sum "as is" so it is recognized as a polynomial?

is not possible. For example, you may think that x^n-1 and x^n+1 are both polynomials and therefore you could try to evaluate Resultant[x^n+1,x^n-1,x]. The error message you get

... Resultant::poly2: 1+x^n is not a well-formed polynomial in x.

is because a polynomial in x for this purpose must have a degree which evaluates to an explicit integer and n is a symbol with no value and does not evaluate to an explicit integer.

What does work is something such as

Resultant[x^3 + b, x^2 + a, x]
(* a^3 + b^2 *)
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The automatic Simplify is using the trivial school algebra formula

     Sum[x^i,{i,0,n}]== (x^(n+1)-1)/(x-1)
     Sum[(i+1)x^i ,{i,0,n} ] == D[Sum[x^(i+1),{i,0,n}],x]

and represents it in the shortest form.

Your first term may stay evaluated but un-simplified for any integer n as

   p[x_, n_] :=Append[ Plus @@ Array[ # x^# &, n],\[Ellipsis]]

   p[x, 3]

$$ x+ 2 x^2 + 3 x^3 +\ldots$$

Here the dots prevent evaluations. You can freeze expressions in a preferred form by Inactivate for manipulations or by HoldForm for printing purposes.

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  • $\begingroup$ Unfortunately, this does not really answer the question asked, but nice try. $\endgroup$
    – Somos
    Jul 11, 2023 at 21:54
  • $\begingroup$ Fortuntely, such questions about trying to prevent a CAS doing its job, demand some fantasy. $\endgroup$
    – Roland F
    Jul 12, 2023 at 7:46

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