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Today I was just thinking very trivially that polynomials with integer coefficients give integer values for integer inputs, so they transform $\Bbb Z\to \Bbb Z$ (and $\Bbb Q\to\Bbb Q$ for rationals).

Now, this constrains the shape of the polynomials thoroughly as it is stuck to certain quantised lattice points that it needs to pass through. Geometrically at least it means derivatives, and thus roots etc can only belong to a finite set of possibilities which are partially constrained by the fact that the polynomial will need to be n-derivable (its $(n+1)-$th derivative will have to be $0$) and at the same time it will need to pass through an integer “lattice”.

Now this may seem silly but I was wondering if this additional constraint of $\Bbb Z\to \Bbb Z$ mapping can be used to strengthen certain theorems, or if it’s rather a tautological consequence and thus cannot be used to strengthen anything.

Thanks for any insights or pointers to specific domains/theorems in this area.

buddhabrot
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    there are polynomials with rational coefficients which always give integer values btw, this is when you use binomial coefficients in x instead of x^n –  Sep 05 '20 at 19:02
  • In general it's hard to strengthen a theorem by adding additional structure. Since $\mathbb{Q}$ is a field theorems proven in $\mathbb{Z}$ will be stronger since they are merely a ring, lacking multiplicative inverses. Some theorems, like Eisenstein's Criteria, show that in certain cases proving a result over the integers will automatically imply the result over the rationals. Since the integers are subset of the rationals, results about the rationals automatically include the integers. – CyclotomicField Sep 05 '20 at 19:11

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This is a vague and broad question so I'll interpret it vaguely and broadly. Here are some nice facts about polynomials $f(x) \in \mathbb{Z}[x]$ that aren't true of more general polynomials, that can be used to prove some stuff:

Fact 1: $a - b \mid f(a) - f(b)$.

For example, $f(1) \equiv f(3) \bmod 2$ and so forth. This is an easy observation but it's a strong and useful constraint on the values that an integer polynomial can take, and in particular it shows that unlike over a field, we can't make an integer polynomial take arbitrary integer values at a finite set of points. It can be used to give a Euclid-style proof of the fact that the set of prime divisors of the values of $f$ must be infinite, which in turn can be used to prove, for example, that the set of primes congruent to $1 \bmod 4$ is infinite, by taking $f(x) = x^2 + 1$.

Fact 2: If $f(x)$ is monic and has nonzero constant term then it has at least one complex root of absolute value at least $1$. Hence if you can show that $f(x)$ has exactly one such root then it must be irreducible.

This can be used to prove Perron's criterion, and a variant of this argument was used on MathOverflow (due to Bjorn Poonen going off of an observation I made) to prove that the polynomials $x^n + p_1 x^{n-1} + \dots p_{n-1}$ are always irreducible, where $p_i$ are the primes! A similar argument also proves this irreducibility criterion which is sort of a mix of the previous two and which doesn't have a name that I know of.

Fact 3: If $f(x)$ is monic and has constant term $\pm 1$ then either it has at least one complex root of absolute value greater than $1$ and another of absolute value less than $1$, or all of its roots have absolute value exactly $1$.

In the latter case it turns out that all of the roots must in fact be roots of unity (so $f$ must be a product of copies of $x$ and the cyclotomic polynomials); this is due to Kronecker and you can find the proof here, which crucially relies on the observation that there are finitely many integer polynomials of a given degree with a given bound on the size of the coefficients. I use this same observation here to show that there exists a nontrivial lower bound on the absolute value of the largest root (in absolute value) of a monic polynomial $f(x) \in \mathbb{Z}[x]$ with constant term $\pm 1$.

In general using the discreteness of the integers is a very common proof technique and there are many examples although it's hard to search for them at that level of generality.

Qiaochu Yuan
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