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.