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.

$$ \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $$

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

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?

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 $a \in \mathbb{Z}$, this norm is a rational quadratic form over the integers with rational coefficients.

$$ \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $$

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

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.

$$ \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $$

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