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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.

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 $\displaystyle |f \left( x_1, x_2, x_3, x_4 \right)| \geq \gamma~~ \text{for all} ~~ x_1, x_2, x_3, x_4 \in \mathbb{Z}$ ?

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.

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What does it mean for a polynomial in more than one variable to have real roots? –  Qiaochu Yuan Mar 15 '12 at 22:42
    
It means that there exist real numbers $x_1^*, x_2^*, x_3^*, x_4^*$ such that $f(x_1^*, x_2^*, x_3^*, x_4^*) = 0$. –  Stewart Mar 15 '12 at 22:45

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up vote 3 down vote accepted

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.

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another related theorem - the fractional parts of $\alpha n^2$ are equidistributed in the unit interval (by a refinemnet of the Kronecker lemma, proved by Weyl/Hardy, and by Frustenberg). I'm guessing that if you take the coeff. if $f$ to be drwan in some iid manner and to be say diophantine generic, you can even get a quantitative version of those results, that is to get estimate on the maximal sizes of $x_1,\ldots, x_4$ which are needed in order to get $f(x_{1},\ldots,x_{4})$ to be less than some epsilon. –  Asaf Mar 15 '12 at 23:34
    
Thanks for the pointer GH, I am already looking into it! Do you maybe know if there is an analogue for higher degree polynomials as well? –  Stewart Mar 15 '12 at 23:40
    
@Stewart: I think there is an analogue for higher degree polynomials as long as the number of variables is large (in terms of the degree). Traditionally these problems are treated by the Hardy-Littlewood circle method (even when the variables are restricted to special integers such as primes). I am sure MathSciNet or the web helps you find some interesting literature. Margulis' achievement in the quadratic case was to bring down the number of variables to 3. –  GH from MO Mar 15 '12 at 23:43
    
@GH: My problem in its full generality involves $n$-degree polynomials with $n^2$ variables. So I think I will be able to show something there. Your suggestions seem to be critical, thank you very much. –  Stewart Mar 15 '12 at 23:57

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