On Oct 25, 2007, at 4:23 PM, Martin Albrecht wrote:

> On Friday 26 October 2007, Robert Bradshaw wrote:
>> This is an interesting construction, but I am wondering if a uniform
>> distribution for all polynomials of specified degree < d, with a
>> specified number of terms, is the most natural one to give, and how
>> grave the impact is on efficiency. (Depending on the coefficient
>> ring, this goal may not even be achievable).
>
> Yes, a random boolean multivariate polynomial will have $MM/2  
> monomials i.e.
> 1/2 of all possible monomials. That is okay, we are not enforcing  
> #T to be
> achieved but we are enforcing that the coefficients determine if we  
> get to #T
> or not rather than the implementation.

For boolean multivariate polynomials, it makes sense to talk about  
the number of terms of a random polynomial of a given degree, because  
you are working with a finite space, but this has a very different  
feel than a random polynomial over an infinite ring, say, Z.

> Actually, Steffen jokingly proposed earlier to have .some_element()
> besides .random_element() which gives you a somewhat random element
> whereas .random_element() actually gives you something very random.

The random_element() function should probably take arguments  
specifying the distribution--some_element makes it sound like it's  
always going to give the same thing.

>> Also, rather than specifying maximum degree/number of terms, it might
>> make more sense (and be much after to implement) to use a
>> distribution with an expected degree and/or number of terms.
>
> can you elaborate on this, I don't really understand that point.

My point is that just because one isn't using a uniform distribution  
doesn't mean that the polynomials produced are any less random. For  
instance, fix a degree d and a (real) distribution X with support  
[0,infinity) and expected value 1. Now let r_1, ..., r_n be chosen  
out of X and let

f = sum x_i ^ round(d/n * r_i)

Now f is a monomial with expected (i.e. average) degree d (I think-- 
not sure how bad the discontinuous rounding would mess things up),  
though it may have degree more or less than d, and though it is not  
uniformly chosen from any set is not any less "random." It can also  
be computed very quickly.

- Robert


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