What is the a name for functions of the form $f(a_1/b_1,\ldots,a_n/b_n) = c/d$ where $a_1,\ldots,a_n,b_1,\ldots,b_n,c,d \in Z$ and all the denominators are not zero.

I was thinking about calling them rational functions; but that name is already taken :).

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    $\begingroup$ Maps $\mathbf{Q}^n\to\mathbf{Q}$. $\endgroup$ – Unit Sep 1 '15 at 5:35
  • $\begingroup$ Thanks, this is a clearer notation. I would like to know whether there is an English name for these functions. $\endgroup$ – Willem Sep 1 '15 at 5:54
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    $\begingroup$ What special properties differentiate your function from $\mathbb{R}^n \rightarrow \mathbb{R}$, and how are they going to be useful? and finally is it called Rational Number Function? $\endgroup$ – Ashutosh Gupta Sep 1 '15 at 5:59
  • $\begingroup$ Thank you. I will take that as an answer. I am no mathematician; but I was thinking about the problem of finding optimal values for such functions (e.g. for which arguments is f at its highest value). [The optimal value can be guaranteed to be found - assuming there bounds placed on the arguments] I assumed there is lots of work done in this area; I just did not know how to find it. So I asked the question here. $\endgroup$ – Willem Sep 1 '15 at 6:17
  • $\begingroup$ Your answer helped already ... gmplib.org/manual/Rational-Number-Functions.html $\endgroup$ – Willem Sep 1 '15 at 6:19

The restriction to rational (instead of real) arguments will just make any treatment harder. As the rationals are dense, it can very well be that your function has a maximum in the reals (e.g. at $\sqrt{2}$), but none in $\mathbb{Q}$, and for practical purposes it makes no difference.

Just consider that e.g. linear programming, a widely used and well studied problem. For real solutions there are efficient solution methods which are able to handle huge problems, if you restrict (some of the variables) to be integers, there aren't really any techniques capable of handling more than modest sized ones.

  • $\begingroup$ You are correct - for practical purposes there are no differences. A rational number function cannot have an answer of $\sqrt{2}$ by definition. However a rational number function may have something that approximates that value. I am thinking about rational number functions are not well suited to linear programming; where the relationships are not linear; and the derivatives are not known: en.wikipedia.org/wiki/Derivative-free_optimization $\endgroup$ – Willem Sep 1 '15 at 16:33

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