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Let $f(x)=(x-a_1)(x-a_2)\cdots(x-a_m)$ and $g(x)=(x-a_{m+1})(x-a_{m+2})\cdots(x-a_n)$, where $m,n \in \mathbb{N}$, $m < n$, $a_i \in F$, $F$ is a field of characteristic zero (for example, $k \in \{\mathbb{R},\mathbb{C}\}$). Let $\lambda,\mu \in F$.

How to compute the subresultants of $f(x)-\lambda$ and $g(x)-\mu$?

If, for example, $m=2$ and $n=4$, then I have succeeded to compute it using $a,b,c,d$:

subresultants [(x-a)(x-b) -\lambda, (x-c)(x-d) - \mu].

But it seems that beyond $a,b,c,d,e$ it is not possible? Or just because I have written $f,g$ (which probably denotes functions). Writing the $a_i$'s does not work.

Thank you very much!

user237522
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    It's not clear exactly what you're asking here; but note that WolframAlpha will happily treat expressions like a[1], a[2], ... the same way it treats undefined constants like a, b, .... For example, the expression Product[ x-a[j] , {j, 1, m} ] will generate the general monic degree-$m$ polynomial $(x-a_1)(x-a_2)\cdots(x-a_m)$. – Greg Martin Dec 29 '18 at 19:53
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    I don't have any trouble w/ command like Subresultants[(x-a)*(x-b)*(x-c)-\lambda,(x-d)*(x-e)*(x-f)-\mu,x]. – achille hui Dec 29 '18 at 19:55
  • Thank you, @GregMartin. Sorry for not being clear. However, you understood what I meant. – user237522 Dec 29 '18 at 19:55
  • @achillehui, thank you. Perhaps the problem was that I have not used * – user237522 Dec 29 '18 at 19:58
  • Something like Subresultants[Product[x-a[i],{i,1,2}]-\lambda,Product[x-b[j],{j,1,3}]-\mu,x]. Replacing upper bound of $i$ by $3$ doesn't work. Replacing upper bound of $j$ to $4$ work and to $5$ doesn't work. It looks like an arbitrary limit WA imposed on its online computational capacity. In order words, a hint to suggest you to use Mathematica instead. – achille hui Dec 29 '18 at 20:15
  • @achillehui, your example is very helpful, thank you. I have tried: Subresultants $(x-a[1])(x-a[2])(x-a[3])-\lambda, (x-b[1])(x-b[2])(x-b[3])-\mu$, but got an error.. as you mention. Is it possible to compute this in Mathematica? – user237522 Dec 29 '18 at 20:26
  • @user237522 Sorry, I don't have access to Mathematica. – achille hui Dec 30 '18 at 01:41
  • @achillehui, ok, thank you for your help. Please see https://mathematica.stackexchange.com/questions/188570/computing-subresultants/188571?noredirect=1#comment491663_188571 – user237522 Dec 30 '18 at 01:45
  • @achillehui Ok, I ran the code snippet from your comment yesterday on my copy of Mathematica, and taking {i,1,3} and {j,1,5} simultaneously took about 20 seconds on my laptop. So I agree with you—probably an imposed limit of WA. – Greg Martin Dec 30 '18 at 21:09

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