User:Chenxlee/AO

In mathematics, a Diophantine inequality is an indeterminate polynomial inequality that requires the variables to be integers. Their study forms a part of number theory and more particularly Diophantine approximation, although methods from the geometry of numbers and, recently, ergodic theory have also been used.

Examples
A general Diophantine inequality is of the form
 * $$Q(x_1,\ldots,x_n)>0,\,$$

for a polynomial Q with real coefficients. The fact that this inequality is Diophantine refers to the fact that an integer solution is desired rather than simply a real one. The condition that the coefficients of the polynomial be real cannot be relaxed to them being complex, since there is no ordering on the complex numbers. They can however be generalised to belonging to a real closed field, and since all such fields have characteristic zero and so contain a ring isomorphic to the integers, the consideration of integer solutions still holds.

Usually of more interest than a single Diophantine inequality is the study of a system of them, that is finding simultaneous integer solutions to a finite collection of Diophantine inequalities
 * $$\begin{align}&Q_1(x_1,\ldots,x_n)>0\\&\vdots\\&Q_n(x_1,\ldots,x_n)>0.\end{align}$$

Another variation on the study of a single Diophantine inequality that is often considered is the study of inequalities involving polynomials in the absolute values of the variables, or even involving the absolute value of polynomials, such as inequalities of the form
 * $$|x_1|+|x_2|+\ldots+|x_n|\leq\lambda\,$$

or
 * $$|x_1x_2\cdots x_n|\leq\lambda\,$$

for some real number λ.

Questions
The questions asked concerning Diophantine inequalities are usually of an existential nature, that is: "do solutions exist?" rather than "what are all the solutions?" Depending on the underlying polynomials certain solutions may be implicitly ignored. A common kind of Diophantine inequality studied are those of the form
 * $$|F(x_1,\ldots,x_n)|\leq\lambda\,$$

f . For example, Minkowski proved that the Diophantine inequality
 * $$|x_1|+\ldots+|x_n|\leq\sqrt[n]{n!}$$