chapter  12
13 Pages

## On the Ideal Generated by the Values of a Polynomial

ByJean-Luc Chabert, Sabine Evrard

In this introduction we assume that f is primitive, that is, C(f) = D. When f ∈ Z[X ], it is well-known that the gcd of the values of f on Z divides n!. For Dedekind domains, this result was generalized by Po´lya [8, S4] in the following way (see also [5, II.3.3]): the ideal D(f) divides the nth factorial ideal n!D where n!D is deﬁned by

n!D = ∏

with N(m) = Card(D/m) (12.3)

and

wq(n) = ∑ l≥1

[ n

ql

] . (12.4)

Writing the ideal D(f) in the following form

D(f) = ∏

m∈max(D) mdm(f), (12.5)

this divisibility relation may be written as inequalities. For each maximal ideal m of D, one has:

dm(f) ≤ wN(m)(n). (12.6)

The aim of this chapter is to state another divisibility relation making use of the number of coeﬃcients of f not belonging to m instead of the degree of f . More precisely, let

µm(f) = Card {ak | ak /∈ m}; (12.7) we are going to prove that

dm(f) < µm(f). (12.8)

But this inequality holds only for small values of n = deg(f), namely:

deg(f) ≤ char(D/m)× (N(m)− 1). (12.9)

Vaˆjaˆitu [9, Theorem 2] proved Inequality (12.8) when D = Z. Here, we generalize it to every Dedekind domain D (Proposition 4.1). Then, we extend it to the ideal D(f,E) generated by the values of f on a subset E of D when D = V is a discrete valuation domain (Proposition 5.4).