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I'm interested in many things, but algebraic geometry I love.
Feb 12 |
comment |
Are there n polynomials for which all intersection multiplicities are at least m?
Look at the discussion above -- what you suggest will only make the condition hold at $x=a$, but not for other values of $x$. For example try multiplying $0,x,1,x+1$ (which satisfy the condition over $x=0$) by $(x-3)^2$ and see that there are values where some of these intersect with multiplicity $1$. |
Feb 12 |
comment |
Are there n polynomials for which all intersection multiplicities are at least m?
@Will: I want that all of its nonzero roots will have multiplicity $\geq m$. Furthermore, I want that for every $i$ there will exist a unique $j$ such that $0$ is a root of $f_i-f_j$. |
Feb 12 |
revised |
Are there n polynomials for which all intersection multiplicities are at least m?
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Feb 12 |
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Are there n polynomials for which all intersection multiplicities are at least m?
You're right! So I guess this proves the $m=2$, $n=4$ case. I can't see how this would generalize, though... |
Feb 12 |
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Are there n polynomials for which all intersection multiplicities are at least m?
Let's do a quick example: $0,x,1,1+x$ have the desired property at 0. If $m=2$, you're saying to multiply by $(x-1)^2$, say. $1(x-1)^2$ and $x(x-1)^2$ indeed intersect with multiplicity $2$ at $x=1$, but they would also intersect with multiplicity $1$ at some $x\neq 0,1$. So that's undesirable. |
Feb 12 |
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Are there n polynomials for which all intersection multiplicities are at least m?
@Mahdi: I don't see why that would solve it. It would just change the intersection number over $x=a$, but not over other values of $x$. |
Feb 12 |
revised |
Are there n polynomials for which all intersection multiplicities are at least m?
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Feb 12 |
comment |
Are there n polynomials for which all intersection multiplicities are at least m?
Hmmm, let me clarify in the body of the question. |
Feb 12 |
asked | Are there n polynomials for which all intersection multiplicities are at least m? |
Feb 4 |
answered | Why should the anabelian geometry conjectures be true? |
Jan 25 |
asked | What is the obstruction for a local set of models of a curve to come from a global model? |
Dec 25 |
awarded | Popular Question |
Dec 11 |
accepted | Where was Riemann Existence first proven? |
Dec 10 |
comment |
Where was Riemann Existence first proven?
I meant the following: any topological cover of an algebraic variety defined over the complex numbers can be given an algebraic variety structure such that the covering map is algebraic. |
Dec 10 |
asked | Where was Riemann Existence first proven? |
Nov 24 |
comment |
General cohomology groups and motives
@Daniel: While I agree that this seems to have no clear answer and is therefore worthy of closing down, I completely disagree that MO should not be the first place to go to with this sort of question. It is precisely the collective knowledge of mathematicians that I want to tap into. There is clearly no book about this; if there are papers about it they would be hard to find; and if individuals don't know the answer it doesn't mean that an answer doesn't exist. This is precisely what mathoverflow is for. |
Nov 24 |
comment |
General cohomology groups and motives
@Daniel: Alas, I did not. My understanding of Langlands is rudimentary at best. I was trying to place Langlands, which is a statement about Galois actions on certain groups coming from schemes, in the context of general Galois actions on groups coming from schemes. |
Nov 24 |
comment |
General cohomology groups and motives
@Daniel: that's a very good point. Does $H^i(X,\mathbb{Q}_p)$ being automorphic imply anything of any content about the representation $H^i(X,\mathbb{Z}/p^n\mathbb{Z})$? |
Nov 23 |
comment |
General cohomology groups and motives
I'm not sure if lisse l-adic sheaf is really what I would be looking for. Ideally, I would like a statement that would also make sense for $\mathcal{F}$ a linear algebraic group scheme (which would restrict us to non-abelian cohomology), although that might be asking too much. For now it would suffice to restrict ourselves to sheaves into abelian groups. |
Nov 23 |
comment |
General cohomology groups and motives
Hmm... I was careless with my speech. You are right that I meant $\mathcal{F}$ as sheaf on the etale site on $X$ rather than $X$. (I viewed that as implicit by the fact that I look at $H_{et}^*(X,\mathcal{F})$.) Your other comment is more substantive -- is it true that $H^i_{et}(X,\mathbb{Q}_p)$ where $\mathbb{Q}_p$ is the constant sheaf on the etale site over $X$ is different from the inverse limit of the $H^i_{et}(X,\mathbb{Z}/p^i\mathbb{Z})$? |