Timeline for Quartic case of a theorem of Bombieri and Pila
Current License: CC BY-SA 3.0
4 events
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| Jun 29, 2011 at 12:53 | history | edited | Stanley Yao Xiao | CC BY-SA 3.0 |
added 205 characters in body
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| Jun 29, 2011 at 12:52 | comment | added | Stanley Yao Xiao | @ACL: Theorem 15 isn't what I am looking for exactly, since it furnishes a positive integer $D$ and $k$, where $k$ is bounded in terms of the products of the $B_i$'s, $T$, and $\lVert F \rVert$, such that every tuple counted by $N(F,B)$ (which implicitly depends on $k$) is a zero of one of $k$ polynomials $F_1, \cdots, F_k$. However, for the bound on $N(F,B)$ to be effective, there has to be some control over the number of zeroes of $F_1, \cdots, F_k$, which is provided by the Bombieri and Pila theorem when the hypotheses of that theorem hold. | |
| Jun 28, 2011 at 22:59 | comment | added | ACL | Theorem 15 of that paper is probably what you need. The superfluous logarithmic term may be removed via the hint given just before Section 2.6: "after treating the factor $(\log \| F\|)^5$ through a version of the process employed for Theorem 5. | |
| Jun 28, 2011 at 18:30 | history | asked | Stanley Yao Xiao | CC BY-SA 3.0 |