Timeline for Infinitely many initial ideals for non-Artinian monomial orders?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 23, 2021 at 9:33 | vote | accept | Igor Makhlin | ||
| Jan 21, 2021 at 0:43 | answer | added | Gjergji Zaimi | timeline score: 3 | |
| Dec 1, 2019 at 1:47 | comment | added | Igor Makhlin | @GjergjiZaimi, thanks! Maybe I should've written that explicitly in my question but the homogeneous case is "easy": if the ideal is homogeneous and has finite-dimensional homogeneous components with respect to some grading of the variables, then virtually any argument for the Artinian case applies (for instance, the one given by Sturmfels). The tricky part is treating ideals for which no such grading exists. | |
| Dec 1, 2019 at 0:51 | comment | added | Gjergji Zaimi | A reference for the homogeneous case is proposition 2.1 in arxiv.org/abs/1512.02662 | |
| Oct 16, 2019 at 16:10 | history | edited | Igor Makhlin | CC BY-SA 4.0 |
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| S Sep 22, 2019 at 21:02 | history | bounty ended | CommunityBot | ||
| S Sep 22, 2019 at 21:02 | history | notice removed | CommunityBot | ||
| S Sep 14, 2019 at 19:10 | history | bounty started | Igor Makhlin | ||
| S Sep 14, 2019 at 19:10 | history | notice added | Igor Makhlin | Draw attention | |
| Apr 30, 2019 at 0:25 | comment | added | Igor Makhlin | Somewhat unexpectedly for myself I found a simple argument showing that the set of initial ideals is indeed finite when $n=2$. I still have no idea what happens when $n\ge 3$, however. | |
| Apr 5, 2019 at 19:47 | comment | added | Sam Hopkins | it's clear now, thanks. | |
| Apr 5, 2019 at 19:47 | history | edited | Igor Makhlin | CC BY-SA 4.0 |
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| Apr 5, 2019 at 19:44 | comment | added | Igor Makhlin | @SamHopkins Yes, that is exactly what I mean. I thought this would be clear from the overall setting but I guess I better reword it if it isn't. | |
| Apr 5, 2019 at 19:40 | comment | added | Sam Hopkins | Are you saying, as we vary $<$, can we get infinitely many different initial ideals $\mathrm{in}_{<}(I)$? | |
| Apr 5, 2019 at 19:38 | comment | added | Sam Hopkins | What does "may the set of $\mathrm{in}_{<}(I)$ [...] be infinite for an ideal $I$" mean? The way you have defined it, $\mathrm{in}_{<}(I)$ is an ideal. | |
| Apr 5, 2019 at 19:32 | history | edited | Igor Makhlin | CC BY-SA 4.0 |
I made the question more reader-friendly to attract more interest. Because, frankly, I believe this to be a very natural question which turns out to be unexpectedly difficult and, therefore, deserves more attention.
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| Apr 5, 2019 at 1:03 | history | asked | Igor Makhlin | CC BY-SA 4.0 |