Parametrizing over $\Z$ integral values of polynomials over $\Q$

Giulio Peruginelli (1) , Umberto Zannier (2)

1 Institut fur Mathematik
2 SNS - Scuola Normale Superiore di Pisa

Given a polynomial $f(X)$ with rational coefficients we investigate the conditions under which the values attained by $f(X)$ over the integers are parametrizable by a polynomial with integer coefficients in one or possibly several variables. That is, given $f\in\Q[X]$ we look for a polynomial $g\in\Z[X_1,\ldots,X_m]$, for some $m\in\N$, such that $f(\Z)=g(\Z^m)$. Obviously, such a polynomial $f(X)$ has to be integer-valued, that is $f(\Z)\subset\Z$. Using Hilbert Irreducibility Theorem, we give an exhaustive classification of such polynomials $f(X)$: they are of the form $F(sX(sX-r))/2$, for some $F\in\Z[X]$ and $s,r$ coprime odd integers and $s$ a prime power (possibly equal to $1$). In particular, only powers of $2$ can appear in the common denominator of the coefficients of $f(X)$ and $f(X)$ satisfies the equation $f(X)=f(-X+r/s)$. Moreover, if such condition holds, we can parametrize $f(\Z)$ with a polynomial with integer coefficients in $1$ variable if $s=1$ or $2$ variables otherwise. We also give a criterium which establishes whether a polynomial with rational coefficients is of the above form.

Mots clés: Gauss norm; Image of a polynomial; Integer-valued polynomial; Polynomial mapping; Polynomial parametrization

Domaines: Théorie des nombres [math.NT]

Giulio Peruginelli, Umberto Zannier. Parametrizing over $\Z$ integral values of polynomials over $\Q$. Communications in Algebra, 2010, 38 (1), pp.119-130. ⟨10.1080/00927870902855564⟩. ⟨hal-00795644⟩