Question

I was asked the following question by a colleague and was embarrassed not to know the answer.

Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, the elements of $\mathbb{Z}$ which are linear combinations of $f(x), g(x)$ with coefficients in $\mathbb{Z}[x]$. Then $I$ is clearly an ideal in $\mathbb{Z}$. Let $D>0$ be a generator of this ideal. The question is: what is $D$?

Now, we do have the standard resultant $R$ of $f,g$, which under our hypotheses, is a non-zero integer. We know that $R \in I$ and it's not hard to show that a prime divides $R$ if and only if it divides $D$. I thought $R = \pm D$ but examples show that this is not the case.

Answer 1

This quantity $D$ is known as the "congruence number" or "reduced resultant" of the polynomials f and g. I first saw this in a preprint by Wiese and Taixes i Ventosa, http://arxiv.org/abs/0909.2724. They ascribe the concept to a paper which I don't have a copy of:

M. Pohst. A note on index divisors. In Computational number theory (Debrecen, 1989), 173– 182, de Gruyter, Berlin, 1991.

Answer 2

$D=D(f,g)$ is not particularly well-behaved, is it. For example it's not multiplicative in the variables: if $g=x^2+1$ (nothing special about this example, I'm just fixing ideas) then $D(f,g)$ is the intersection of $\mathbf{Z}$ and the ideal generated by $f(i)$ in $\mathbf{Z}[i]$. So, for example, if $f$ is $x+2$ you get 5, if $f$ is $x-2$ you get 5 too (from the other prime ideal above 5) and if $f$ is $x^2-4$, the product, you still only get 5 (which of course provides a proof that $D$ isn't the resultant in general).

Seems to me that the norm of $f(i)$ would be a much better invariant, which would generalise to the size of the ring $\mathbf{Z}[x]/(f,g)$. It wouldn't surprise me if that were the resultant (or closely related to it), and if it is then that's at least some sort of link. Maybe you knew all that already though ;-)

Answer 3

An interesting example where the resultant and the "reduced resultant" differ comes from the theory of elliptic curves. Take an elliptic curve $$E:\qquad y^2=x^3+ax+b$$ where $a$ and $b$ are integers. The duplication formula for $E$ states that on the elliptic curve $[2] (x_1,y_1)=(x_2,y_2)$ where $x_2=g(x_1)/4f(x_1)$, $$f(x)=x^3+ax+b\qquad{\rm and}\qquad g(x)=x^4-2ax^2-8bx+a^2.$$ The resultant of $f$ and $g$ is $(4a^3+27b^2)^2$ (the square of the discriminant of $E$ so nonzero). But the reduced resultant is $|4a^3+27b^2|$ which one sees by noting that this divides all entries of the adjugate of the "resultant matrix" of $f$ and $g$.

The fact that this resultant is nonzero is used in a standard proof of the inequality $$h([2] P)\ge 4 h(P)-O(1)$$ for the naive height on $E$ (see Silverman's book).

Answer 4

This issue came to my attention for the first (and only, up until now) time when Gregory Dresden gave a talk about resultants of cyclotomic polynomials in the UGA number theory seminar last spring. I am pretty sure that the distinction between these two notions of the resultant figured prominently in his work: see

http://home.wlu.edu/~dresdeng/papers/Res.pdf

So far as I know the question you ask is unsolved in general, but in this case I don't know very far at all. You might want to ask Greg...