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I am reading the paper Dual-to-Kernel Learning with Ideals. Here is part of it:

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The definition/motivation of genericity in Wikipedia are

A generic point of the topological space $X$ is a point $P$ whose closure is all of $X$, that is, a point that is dense in $X$. The terminology arises from the case of the Zariski topology of algebraic varieties. For example having a generic point is a criterion to be an irreducible set.

I cannot see why this gives the linear independence in Theorem 1 immediately. Any hints? Thanks.

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This isn't the meaning of "generic" you're looking for, although there's a historical relationship. If you try to think of your $y_i$ as generic points, or as constituting a generic point, you run into immediate problems since $\mathbb{K}^n$ will not have a generic point under any common topology (points are closed in all important examples.)

The "generic" here means "in general position," i.e. the $y_1...y_m$ aren't supposed to lie on the zero set of any polynomial of degree less than $m-1$. (It's best to think of this for $\mathbb{K}=\mathbb{C}$ and $n=1$.) The claim then is just that being linearly dependent under $k_*$ would give such a polynomial relation.

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