Timeline for Multiplicity of a polynomial in positive characteristic
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21, 2019 at 14:30 | comment | added | bog | Can you elaborate more on "divided powers derivatives"? | |
| Aug 21, 2019 at 5:39 | comment | added | Bugs Bunny | BTW, $p$-th derivativie is always zero in characteristic $p$ :-)) You oughta use divided powers derivatives! | |
| Aug 21, 2019 at 5:37 | comment | added | Bugs Bunny | So, is it just the degree of the first non-vanishing homogeneous component, when you write $f$ as a polynomial in $y_k= x_k-q_k$? Why on Earth do you need the Grobner basis in this case? It looks like a straightforward change of variable calculation,,,, | |
| Aug 21, 2019 at 1:53 | comment | added | Mohan | In positive characteristics, derivatives are often ill-behaved. For example, $x^p-y^{p+1}$ has multiplicity $p>0$, but its $p$th derivatives at the origin is zero. | |
| Aug 20, 2019 at 18:55 | comment | added | Bugs Bunny | What is multiplicity? I understand it with respect to a divisor but not with respect to a point... | |
| Aug 20, 2019 at 18:54 | comment | added | bog | Thank you. I was looking for a linear algebra method, like in characteristic 0 case, where it suffice to take partial derivatives. Is there some analogue in positive characteristic? | |
| Aug 20, 2019 at 16:17 | comment | added | Mohan | One way (how good computationally, this is, I do not know) is to change $x_i$'s to $x_i-q_i$ where $q=(q_i)$ and thus assume $q$ is the origin. Then the multiplicity is the least degree of $f$, that is, we can write $f=f_p+f_{p+1}+\cdots$ where $f_k$ is the homogeneous term of degree $k$ and $f_p\neq 0$. The the multiplicity is $p$. | |
| Aug 20, 2019 at 15:42 | history | asked | bog | CC BY-SA 4.0 |