Timeline for polynomials in many variables and Hasse principle
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2014 at 16:19 | answer | added | Bjorn Poonen | timeline score: 10 | |
| Sep 28, 2014 at 7:38 | comment | added | Daniel Loughran | Yes rational solutions are much easier. This is because for such problems, the existence of a rational solution to a polynomial is equivalent to the existence of a rational solution for its homogenisation. This is of course not true in general, but the key point is that when the circle method works, it shows that if rational solutions exist then there are lots of them, namely that they are Zariski dense. So they can't all lie at infinity, so to say. Here therefore classical work of Birch should give the result you want (with exponential growth, as I explained above). | |
| Sep 27, 2014 at 22:16 | comment | added | user58702 | @DanielLoughran Just out of curiosity, would considering rational solutions instead of integral ones make the problem much easier? Do you know of any results (or similar) in the case of rational solutions? thanks again. | |
| Sep 27, 2014 at 17:42 | comment | added | Daniel Loughran | Unfortunately I don't, though I'm not an expert in the circle method. If you are feeling brave, you should try emailing some circle method experts. Its an interesting problem and I'm sure they would be receptive. | |
| Sep 27, 2014 at 16:39 | comment | added | user58702 | Yes, I've tried looking at the literature a bit, but all seems geared towards forms instead of just polynomials. Do you know any references which might deal with a case similar to mine (i.e. with polys instead of forms)? Thanks! | |
| Sep 27, 2014 at 14:12 | comment | added | Daniel Loughran | My naive guess is that the circle method should be able to give the result you want when $n=4$. However you are right that people normally consider homogeneous forms as they are easier, and I don't know whether the result you want has been worked out. Have you already tried looking at the circle method literature to see if you can find the result you want? | |
| Sep 27, 2014 at 14:09 | comment | added | Daniel Loughran | Ok, this makes the problem quite a bit more difficult. Let me just mention that the answer to your question as stated is no (even in the case of homogeneous forms). Here the circle method is usually used to tackle such problems, and when it works it yields a bound for $k$ which his exponential in the degree $n$. Whereas you seem to be looking for a linear bound on the degree. Such bounds are not known in general. Of course if you are only interested in the case $n=4$, then this is not so much a problem. | |
| Sep 27, 2014 at 13:49 | history | edited | user58702 | CC BY-SA 3.0 |
added 47 characters in body
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| Sep 27, 2014 at 13:49 | comment | added | user58702 | Integral solutions (sorry I forgot to mention it) | |
| Sep 27, 2014 at 12:26 | comment | added | Daniel Loughran | Are you interested in rational solutions or integral solutions? | |
| Sep 27, 2014 at 12:01 | review | First posts | |||
| Sep 27, 2014 at 12:56 | |||||
| Sep 27, 2014 at 11:59 | history | asked | user58702 | CC BY-SA 3.0 |