most recent 30 from http://mathoverflow.net 2013-05-20T19:44:11Z http://mathoverflow.net/feeds/question/91335 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91335/diophantine-approximations-and-quadratic-polynomials Stewart 2012-03-15T22:35:58Z 2012-03-15T23:00:51Z

<p>I am working on a problem these days and the following issue came up. I am not sure yet that I understand it's depth very well, so I would like to discuss a simple case. For those interested, the problem has applications in coding theory. </p> <p>Consider a quadratic polynomial $f \left( x_1, x_2, x_3, x_4 \right)$ with real roots and coefficients drawn from a continuous distribution (and therefore irrationals with probability 1). Is there a strictly positive lower bound on $|f \left( x_1, x_2, x_3, x_4 \right)|$ if we constrain all $x_1, x_2, x_3, x_4$ to lie in $\mathbb{Z}$ ? In other words, is there a $\gamma > 0$ such that $\displaystyle |f \left( x_1, x_2, x_3, x_4 \right)| \geq \gamma~~ \text{for all} ~~ x_1, x_2, x_3, x_4 \in \mathbb{Z}$ ?</p> <p>It seems to me that this is a Diophantine approximation - type problem. Note that one can show through a simple application of Khintchine-Groshev theorem that $|f \left( x_1, x_2, x_3, x_4 \right)|$ will be strictly positive for all $x_1, x_2, x_3, x_4 \in \mathbb{Z}$, in that case with probability 1. This is relatively straightforward. </p>

http://mathoverflow.net/questions/91335/diophantine-approximations-and-quadratic-polynomials/91336#91336 GH 2012-03-15T23:00:51Z 2012-03-15T23:00:51Z

<p>I think the answer is no. For example, the Oppenheim conjecture (proved by Margulis in 1987) states that if an indefinite nondegenerate quadratic form has at least 3 variables and it is not proportional to a rational quadratic form, then its set of values taken at integers are dense in the real line.</p>