Timeline for irreducible elements in a ideal of $R[x_1,x_2]$
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2012 at 15:35 | history | edited | Guntram | CC BY-SA 3.0 |
added 28 characters in body
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| Apr 6, 2012 at 2:28 | answer | added | Will Sawin | timeline score: 7 | |
| Apr 6, 2012 at 0:26 | comment | added | Daniel | But do you mean dimension as a vector space? remember that here the "scalars" can also be polynomials | |
| Apr 6, 2012 at 0:08 | comment | added | Noam D. Elkies | @Daniel: Try mathoverflow.net/questions/88895 (I see that if you click on what I wrote the final "]" gets appended to the URL). What I wrote has a slight typo: should be $N > \#(S) + O(1)$, not $\#(S) = O(1)$. Basically, once $N$ is large enough (bigger than the size of the finite subset plus a bit), there aren't enough reducible polynomials of degree $N$ to fill the space of degree-$N$ polynomials vanishing on $S$. (If $N < \#(S)$ the claim can fail, because all the points in $S$ might lie on one line $l$, and then a degree-$N$ polynomial vanishing on $N$ would vanish identically on $l$.) | |
| Apr 5, 2012 at 23:59 | comment | added | Daniel | I don´t understand your answer, and your link is deleted. | |
| Apr 5, 2012 at 21:06 | comment | added | Noam D. Elkies | Let ${\cal P}_N$ be the space of degree-$N$ polynomials. The reducible polynomials in ${\cal P}_N$ form the union of subvarieties each of codimension at least $N-O(1)$ [see e.g. my answer to mathoverflow.net/questions/88895]. The polynomials vanishing on a finite set $S$ of points constitute a subspace of ${\cal P}_N$ of codimension at most $\#(S)$. Therefore, once $N > \#(S) = O(1)$ most polynomials in that subspace are irreducible, QED. | |
| Apr 5, 2012 at 20:09 | comment | added | Daniel | But how can I prove that it´s irreducible? | |
| Apr 5, 2012 at 20:05 | comment | added | J.C. Ottem | Take $F=\sum_{i+j\le N} a_{ij}x_1^jx_2^j$. To pass through the points imposes a finite set of conditions on the coefficients $a_{ij}$. Taking $N$ sufficiently large you can find such an $F$ which is also irreducible. | |
| Apr 5, 2012 at 19:50 | history | asked | Daniel | CC BY-SA 3.0 |