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Andre Weil noticed as a student in 1925 that the polynomial ring $\mathbb{Z}[x]$ comes close to being a PID, and he describes this as `` the embryo of my future thesis.''

He observed that, given $f(x),g(x)\in\mathbb{Z}[x]$, the Euclidean algorithm computes a sequence of polynomials where each is a linear combination of the preceding ones and which either:

1) Ends in a term which divides the preceding term and thus is something like a GCD for $f$ and $g$; or

2) Ends in an integer $d$ which does not divide the preceding term but bounds the common divisors of values pointwise in this way: For any integer $n$, an integer common divisor of $f(n),g(n)$ must divide $d$.

I have not seen this property discussed anywhere. Does it have a name?

There is related discussion in the question The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$

The number $d$ is called the reduced resultant of $f,g$.

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I do not know of a name. But Weil's observation follows from properties of the resultant, and generalizes to other rings $A[x]$ ($A$ a GCD domain).

The resultant $R(f,g)$ of two polynomials is defined either as a certain determinant or as a certain product over pairs of roots of $f,g$, see this Wikipedia page.

From the two definitions, we see that:

  • The resultant vanishes iff $f,g$ have common root (this is basically the definition of the resultant).
  • The resultant has the property that there exist $p,q \in \mathbb{Z}[x]$ ($\deg p < \deg g$, $\deg q < \deg f$) such that $$(*) p(x)f(x)+q(x)g(x)=R(f,g)$$ identically. In particular, $R(f,g) \in (f(x),g(x))$ - this ideal contains a non-zero integer!

So,

  • If the resultant is 0, there is a genuine GCD.
  • Otherwise, we end up with a non-zero integer $R(f,g)$. Plugging $x=n$ in $(*)$ we see that indeed the GCD of $f(n),g(n)$ divides $R(f,g)$.

The above observations generalize to $A[x]$ when $A$ has the GCD property.

As explained in the wikipedia page, the Euclidean algorithm differs from the resultant calculation by a simple factor.


It is interesting to note that the resultant $R(f,g)$ is not the necessarily the generator of $(f(x),g(x)) \cap \mathbb{Z}$, which we will denote $c(f,g)$. Myerson has shown that $R(f,g)$ divides an (effective) power of $c(f,g)$. When $R(f,g)$ is squarefree and $f,g$ are monics it may be shown that $c(f,g)=R(f,g)$. This may be seen from the results of this recent paper of Frenkel and Pelikán, who study the possibly values of $\gcd(f(n),g(n))$ as $n$ ranges over $\mathbb{Z}$.

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  • $\begingroup$ The relation between $R(f,g)$ and $(f(x),g(x))\cap\mathbb{Z}$ was also discussed in this MO question. $\endgroup$ – Jarek Kuben Jan 29 '17 at 21:41
  • $\begingroup$ Does this use of the resultant generalize to multiple variables? Say $f(x,y,z),g(x,y,z)$. You can calculate the resultant in terms of $x$ by the determinant (treating $y$ and $z$ as parameters). But the ring of polynomials in $y,z$ is not a GCD domain so it is not clear to me what conclusions you can draw about common divisors of values $f(a,b,c),g(a,b,c)$. $\endgroup$ – Colin McLarty Jan 30 '17 at 1:45

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