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The question deals with polynomials in one variable with coefficients in some in some fixed ring $A$.

Given a polynomial $P\in A[X]$, I denote:

  • C(P)= number on non zero coefficients of $P$. (So for example, $C(X)=1$, $C(X^3-4X+2)=3$.)
  • $(P)^*$ the set of non zero multiple of $P$
  • $N(P)=\inf_{M\in (P)^*} C(M)$

So for example, if $P=X^4+X^3+X^2+X+1$, one has $C(P)=5$, but $N(P)=2$ since $P\cdot (X-1)=X^5-1$. So $N(P)=2$ (since it is obviously not $1$).

I am interested in the function $P \mapsto N(P)$. In particular, does it have a name?
And is there a clever way to compute it (say for $A=\mathbf Q$)? If you bound the degree of M, then it is a linear algebra question, but I am not sure how to deal with the fact that $M$ is not bounded.

I guess also that in the generic case (if $A$ is a ring of polynomial whose indeterminates are the coefficients of $P$), this integer has something to do with Hilbert 13th problem, but more precisions on this would be appreciated.

Oblomov
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    Interesting question. Played around with this for a bit but couldn't find a pattern. Maybe start over $A = \Bbb C$ so that at least you know every polynomial factors as a product of linear components.

    Important examples to consider seem to be cyclotomic polynomials. For any integer $n$ we can find a cyclotomic polynomial $P$ such that $C(P) > n$ but $P$ divides $x^m - 1$ for some $m$ so that $N(P) = 2$.

     – Ruben Jun 25 '19 at 14:28
  • There are simple algorithms for recognizing factors of cyclotomics, e.g. see here. – Bill Dubuque Jun 25 '19 at 17:11

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