Timeline for Ideal membership and change of fields
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 21, 2023 at 19:52 | comment | added | Vladimir Dotsenko | @TC what I would do is compute the Gröbner basis over $\mathbb{Q}$ carefully: do not cancel any common factors, and do not divide by anything. Then, once you compute your Gröbner basis, you will have the finite many leading coefficients, and if $p$ is not a divisor of either of them, you are already good because the $\mathbb{Q}$-Gröbner basis with integer coefficients also works modulo such $p$, so you just have finitely many $p$ to check. Avoiding using Gröbner bases means that you'd like to use something very specific about a particular problem, and it is up to you to figure out what it is. | |
| Jul 21, 2023 at 6:10 | comment | added | T C | There should be multiple ways to show $f\in I$ if it is true, using the presentations, for example. However, it seems harder to show that $f\notin I$. I know computers do this by using Groebner basis. But is there a different way? Sometimes Groebner basis is just too big. | |
| Jul 21, 2023 at 6:09 | comment | added | T C | Thank you. I do have one more question following up with your answer: How do one show that $f\in I$ or $f\notin I$ over $\mathbb{Q}$ and $\mathbb{F}_p$ for all $p$? | |
| Jul 21, 2023 at 6:08 | vote | accept | T C | ||
| Jul 21, 2023 at 6:02 | history | answered | Vladimir Dotsenko | CC BY-SA 4.0 |