I don't know whether this is known or not, but I was thinking of the following problem.


Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f\_1,...,f\_n\in \mathbb{C}[x]$, such that all of their intersection numbers are at least $m$?


It is also possible to state an even harder question in a more natural (albeit less basic) way: Are there $n$ homogeneous polynomials in 2 variables for which all of their intersection numbers in $\mathbb{P}^1\_{\mathbb{C}}$ are at least $m$?


EDIT: Sorry, the comments made me realize I wanted a slightly different condition: that for every $j$ there's a unique $k\neq j$ such that $f\_j(0)=f\_k(0)$. (In particular, $n$ is assumed to be even.) I will allow the intersection at $x=0$ to not be of multiplicity $\geq m$, but I will ask that over $x\neq 0$ the intersection multiplicity will be $\geq m$.


### Clarification


The question was posed in a way that algebraic geometers would understand, because I suspect they are most likely to come up with a solution to this question. I wanted to emphasize, however, that intersection multiplicity is something that every high-school student can understand: If $f\_1$ and $f\_2$ are polynomials with coefficients in $x$, and they meet at say $x=3$, then their intersection multiplicity at $x=3$ is the greatest natural number $l$ such that $(x-3)^l$ divides the polynomial $f\_1-f\_2$.



Are there n polynomials for which all intersection multiplicities are at least m?
Ok, so here's the long version of what I already said in the comments:


Start by picking $g\_1,\ldots,g\_n\in \mathbb C[x]$ such that $g\_i\neq g\_j$ whenever $i\neq j$ which satisfy the prescribed condition at $0$ (that's not hard). Since $g\_i-g\_j\neq 0$ for each $i\neq j$, there will be only finitely many $0\neq a\in\mathbb C$ such that $g\_i(a)-g\_j(a)=0$ for some pair $i,j$ with $i\neq j$. Call these values $a\_1,\ldots,a\_k$. Then define 
$$
 f\_i(x) := \left(\prod\_{j=1}^k (x-a\_j)^m\right)\cdot g\_i(x)
$$ 
Those $f\_1,\ldots,f\_n$ have pairwise intersection also in $a\_1,\ldots,a\_k$ (and possibly in $0$), but now with multiplicty at least $2m$ (except in $0$, where the intersection multiplicty is left as it was). The condition at zero is clearly preserved and we didn't add any further intersection points either.


