Let's consider the space $L^2[a,b]$ of functions on the interval and the norm:


$$ ||f(x)||^2 = \int\_a^b |f(x)|^2 \, dx $$


Now what if we consider only polynomials with integer coefficients: $f(x) \in \mathbb{Z}[x]$? 


If we write $f(x) = \sum a\_n x^n$ with each $a\_n \in \mathbb{Z}$, this norm is a rational quadratic form over the integers with rational coefficients. 


$$ \inf\_{f \in \mathbb{Z}[x],f\neq 0} \int\_a^b f(x)^2 \, dx $$


Does it obtain a minimum value over the set of integer-valued polynomials? What is it?



minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $
Thanks to **BigM** for the link to **Ofir's**
[MO Question 121913](https://mathoverflow.net/questions/121913),
which cites a 120-year-old paper of Hilbert for the result that
the integral can get arbitrarily small as long as $b-a < 4$:



> 
> D. Hilbert: Ein Beitrag zur Theorie des Legendre'schen Polynoms, *Acta Math.* **18** (1894), 155$-$159
> 
> 
> 


If $b-a \geq 4$ then an elementary argument using properties of
Legendre orthogonal polynomials shows that there is a positive lower bound.
I gave this argument an hour ago in my answer to the earlier MO question;
I guess it must be an old result, perhaps known already to Hilbert
(who mentions Legendre polynomials in the title of his paper!),
but it's the kind of result that's easier to prove than to find
in the literature. That question did not ask for the value of the minimum,
but I see that the bound is an increasing function of $\deg f$,
and is sharp for $\deg f = 0$, whence the minimum value is $b-a$,
attained only by the constant polynomials $\pm 1$.


