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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?

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    $\begingroup$ You might find this LINK useful. $\endgroup$
    – BigM
    Commented Dec 4, 2014 at 1:38
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    $\begingroup$ Integer valued polynomials is not the same as polynomials with integer coefficients. Which one do you want? $\endgroup$
    – Felipe Voloch
    Commented Dec 4, 2014 at 3:03
  • $\begingroup$ @FelipeVoloch you are right. I have corrected the title $\endgroup$
    – john mangual
    Commented Dec 4, 2014 at 3:10
  • $\begingroup$ But the last line of the question still says "integer-valued". $\endgroup$
    – Gerry Myerson
    Commented Dec 4, 2014 at 5:08
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    $\begingroup$ to me it seems that the answer depends on the values of $a$ and $b$; if both are in the in the open interval $(-1,+1)$, then the infimum is $0$ and no minimum exists. I would therefore suggest to first clarify, how the answer is for $a=0,b\ge 1$ i.e. how it depends on $b$ in that case. $\endgroup$
    – Manfred Weis
    Commented Dec 4, 2014 at 5:55

1 Answer 1

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Thanks to BigM for the link to Ofir's MO Question 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 $\pm 1$.

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