## Transformation of Diophantine equation into another one or a series [closed]

Two questions: First,Given a Diophantine equation $P(y,x_1,\cdot \cdot \cdot,x_j)=0$ where $y \in S,S \subseteq \mathbb{Z}$ is set of the solutions of the Diophantine equation when $x_1,\cdot \cdot \cdot,x_j \in \mathbb{Z}$ ,how much is the number of variables of a Diophantine equation equivalent to it (that is they just have the same set of solutions S of n over $\mathbb{Z}$)with the least number of variables ? 11?or much smaller or bigger?

Second,Given a Diophantine equation $P(y,x_1,\cdot \cdot \cdot,x_j)=0$ where $y \in S,S \subseteq \mathbb{Z}$ is set of the solutions of the Diophantine equation when $x_1,\cdot \cdot \cdot,x_j \in \mathbb{Z}$,is there series in the form $f(y)=\Sigma_0^{\infty}a_i y^i$ where $a_i \in \mathbb{Z}$ or $a_i \in \mathbb{Q}$ equivalent to the diopantine equaton (that is the series have just the same set of solutions (set of zero ) S over $\mathbb{Z}$ as the Diophantine equation S.)?

The second question seems to be true,but I do not know how to prove it.

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I do not know why this question has been voted by one body as unuseful or unclear ,please comment before vote. – XL Jul 14 2011 at 17:51
I was the downvoter. My reasons included much of what David S has pointed out, plus the fact that the title is an incomplete sentence fragment, and the gratuitously random "11". It was not me who voted to close, though... – David Loeffler Jul 14 2011 at 18:33
It is still not clear. It is customary to have something on the right of an equals sign. As you have violated this custom twice, it's incumbent upon you to explain yourself a little better. I vote to close as not a real question. I'm sure there's a real question here somewhere, but it's up to you to shape it up. – Gerry Myerson Jul 15 2011 at 0:55
OK, I now understand the first question. A subset $S$ of $\mathbb{Z}$ is called diophantine if there is a diophantine equation $f(y, x_1, \ldots, x_n)=0$ such that $S$ is the set of $y$ for which this equation is solvable. It follows from work of Matiyasevich and Robinson that there is a universal $n$ such that any diophantine set is universal with that value of $n$, and you want to know what that value is. Wikipedia lists the best bounds as $9$ if the $x_i$ range through positive integers, or $11$ if they range through all integers. (continued) – David Speyer Jul 15 2011 at 3:20
See en.wikipedia.org/wiki/… . The 11 variable paper is available at math.nju.edu.cn/~zwsun/12d.pdf ; I have not read it. There are only 4 papers in mathscinet which cite this one, and none of them improve the bound, so it is likely that this is still the record. – David Speyer Jul 15 2011 at 3:26