5

EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO.

Lately I've been interested in the Hilbert Matrix (its definition will come later). I went on reading Hilbert's original paper on it from 1893.

Hilbert was interested in the following problem: Consider the following inner-product of polynomials: $\langle p, q \rangle = \int_{a}^{b} p(x)q(x) dx$. Can the norm of a non-zero polynomial $p(x) \in \mathbb{Z}[x]$, which is $(\int_{a}^{b} p^2(x) dx)^{1/2}$, get arbitrarily small? In the first 4 pages of his paper he does the the following:

  1. He shows that if $p$ is of degree $<n$ and $v \in \mathbb{Z}^{n}$ is the vector of coefficients of $p$, then this norm is $(\langle v, H_n v \rangle)^{1/2}$ where $H_n$ is the "generalized" Hilbert matrix: $(H_n)_{i,j} = \langle x^{i}, x^{j} \rangle = \frac{b^{i+j+1}-a^{i+j+1}}{i+j+1}, 0 \le i,j \le n-1$. When $a=0,b=1$, this is known as Hilbert matrix. So this problem reduces to that of determining when the following limit is 0: $\lim_{n \to \infty} \min_{0 \neq v \in \mathbb{Z}^{n}} <v,H_n v>$.

  2. Hilbert then uses the orthogonal basis of the inner-product space $\langle f, g \rangle = \int_{0}^{1} f(x) g(x) dx$ ("Shifted Legendre Polynomials"), to calculate $\det H_n$, which is $a_n (\frac{b-a}{4})^{n^2}$, where $\lim a_{n}^{1/n} = $ some positive constant.

On the fifth and last page, Hilbert quotes a paper by Minkowski to show that the answer to his question is affirmative when $b-a < 4$. I found Minkowski's paper here, and it seems that (at page 291) he quotes a result by Hermite that bounds the minimal value of a positive quadratic form over the integers by $n (\det A)^{1/n}$. Using this result and the value of $\det H_n$, Hilbert's result is clear.

My questions are:

1) How is Hermite's\Minkowski's bound on the minimal value of a positive quadratic form proved? Is it a well-known bound? Is there a reference in English?

EDIT: I was able to answer this. Some googling led me to Hermite's constant, which by definition gives the bound $\gamma_n (\det A)^{1/n}$. where $\gamma_n$ is Hermite's constant. It is known that $\gamma_n = \theta(n)$, which is the result I need. More googling gave this survey, and in page 137 Minkowski's bound is proved, and it gives $\gamma_n \le n$.

2) What is known about this problem when $b-a \ge 4$? Is there also a lower bound on the minimal value of a positive quadratic form over the integers?

3) Is there an explicit construction of a sequence of polynomials $p_n$ have a norm tending to 0? (For fixed $a,b$ with $b-a<4$?

Rodrigo de Azevedo
  • 23,223
Ofir
  • 6,343
  • Do you mean $\det H=\lim a_n\left(\frac{b-a}4\right)^{n^2}$? Or, perhaps $\lim (a_n\left(\frac{b-a}4\right)^{n^2})^{1/n}$? This would answer your question, as this one converges to $0$ iff $\frac{b-a}4<1$. – Berci Feb 10 '13 at 22:58
  • This is what I don‘t get: why does small determinant implies small value of the corresponding quadratic form? And is it an "if and only if" implication? – Ofir Feb 11 '13 at 06:27
  • $\det H$ is the product of the $n$ eigenvalues of $H$, counted by multiplicity. If the determinant is $a_n c^{n^2}$, at least one eigenvalue has absolute value $\le (a_n)^{1/n} c^n$, and if $c < 1$ that goes to $0$ as $n \to \infty$. – Robert Israel Feb 11 '13 at 06:41
  • @RobertIsrael But why does the eigenvector has rational or close to rational coordinates? – Ofir Feb 11 '13 at 10:52

0 Answers0