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Let $k$ be an algebraically closed field and let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ variables. Let $d_1,...,d_m$ be non-negative integers. For each $d$, consider the linear subspace $R_d$ of $R$ generated of all monomials of degree $d$.

Given a sequence of polynomials $F=(f_1,...,f_m)\subseteq \prod R_{d_i}$ of degrees $\deg(f_i)=d_i$, we can identify this sequence with a tuple in $k^d$ for some $d$ ($d=\sum_i \dim_k(R_{d_i})$), where the tuple contains the coefficients of each polynomial $f_i$.

I'm trying to understand the concept of generic property:

A property of the sequence of polynomials in $\prod R_{d_i}$ is generic if it holds in a nonempty Zariski-open subset of $k^d$.

According to Keith Pardue, if a property is generic, then this property holds most of the time, that is, that property ought to hold for a randomly chosen sequence.

My question is:

Is this claim a fact? something that you can prove (using statistics tools or whatever)? or it's only a definition based in intuition.

I know that every nonempty open subset of the Zariski topology is dense, which gives some intuition about this. However, Zariski topology is not like Euclidean topology, where you can see that dense subsets contains "almost all" points. Moreover, in $\Bbb R$ for example, being rational (or irrational) is not something you can expect from a random number, however, $\Bbb Q$ is dense in $\Bbb R$ (using euclidean topology).

I've looked for this in a lot of places (M.SE included, of course), but I'm not able to obtain a conclusion. Apparently, this is an intuition fact, but I really don't get the intuition behind it.

It's very important for me to understand why is this. Being a Regular sequences is proven to be a generic property according to the definition I gave, and I really want to know why does this imply that "almost all sequence is a regular sequence". This has huge applications and every reference I see uses this as a fact, but I really don't get it.

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  • $\begingroup$ That a statement holds most of the time means exactly that you should expect that it holds in an arbitrary case, but that there might be some (relatively very few) counterexamples. For instance, if you flip a coin until you get heads, and you do this experiment a lot, most of the time you finish within a finite number of throws. That means that any single time you do the experiment you expect it to not last indefinitely, but there is always the possibility that it might. Zarisky open sets contain, in the same sense, most of the points of your space (unless they're empty). $\endgroup$ – Arthur Mar 30 '16 at 16:30
  • $\begingroup$ @Arthur Thanks for your comment. I get the idea of most of the time, however, I don't see, not even intuitively, why a property most hold most of the time if it holds in a non-empty Zariski open set. Of course, such a property does not hold for some sequences (I'm not saying it's supposed to hold for all the sequences), but it's supposed to hold for almost all sequences, and I really can't see why is this. $\endgroup$ – Daniel Mar 30 '16 at 16:34
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The intuition comes from special fields, endowed with a topology like $\mathbb R$ or $\mathbb C$.
If a property holds for, say, a non-empty open subset of $\mathbb C^n$ the points where it doesn't hold form a closed complex subvariety of $\mathbb C^n$, which is negligible for example because it has Lebesgue measure zero.
However for more exotic algebro-geometric structures, namely schemes, there is no longer such an intuitive interpretation .
For example if $R$ is a discrete valuation ring its spectrum $\operatorname {Spec}(R)$ has only two points, one closed and one open, and the assertion that if you "randomly" choose one of them it will be the open one doesn't seem to make much sense since we dont have a natural probability distribution associated to $\operatorname {Spec}(R)$ .

Conclusion
The phrase
"this property holds most of the time, that is, that property ought to hold for a randomly chosen sequence"
is (in Pauli's cruel formulation) not even wrong, since "randomly chosen sequences" is a meaningless concept for algebraic varieties or schemes.
However the intuition provided is excellent.

Remark
Being both meaningless and intuitively useful is a common apparent paradox : physicists for example have introduced concepts that didn't make sense to their contemporary mathematical colleagues and put them to good use because of their powerful physical intuition.
Illustrations are, for example, Dirac's "function", formalized by Schwartz in the framework of his theory of distributions, and Feynman's integrals which don't seem to have acquired yet a definitive mathematical status.

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I just stumbled across this question. As the guilty author I ought to comment. The question is about my comment in the introduction to my "Generic Sequences of Polynomials" that if a property is generic, then "The property ought to hold for a randomly chosen sequence."

As Georges remarks, my statement is not even wrong. Without more context, it is neither a claim nor a fact nor a definition. At the start of the paragraph in which I make this statement, I write "I use 'random' in a sense that is more evocative than rigorous". As Georges further remarks, the role of the statement is to provide intuition, but I work with the rigorously defined "generic" rather than the evocative term "random" in the rest of the paper.

You can put a statement like this in a context in which it really does have a truth value. In my paper at least, I am not dealing with arbitrary schemes, but with $k^n$ where $k$ is an infinite field. If you put a probability distribution on $k^n$ such that every closed proper subset (with respect to the Zariski topology) is measurable and has measure zero, then it is the case that any generic property will be obtained with probability 1. Your field would need to be uncountable for such a distribution to exist. Given that, since the closed proper subsets have dimension lower than $n$, it would not be unreasonable to require this of your distribution.

You can also talk about generic properties when $k$ is finite, but the statistical interpretation is more subtle and the naive interpretation that I gave can be misleading in this case.

Thanks for engaging with my paper!

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