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I asked a somewhat similar question here, but I believe this one is different enough for its own post.

Let $p : \{1,2 \}^n \rightarrow \mathbb{Z}_3$ be a polynomial chosen uniformly at random (every coefficient is chosen uniformly at random from $\mathbb{Z}_3$). Suppose that $q : \{1,2\}^m \rightarrow \mathbb{Z}_3$ was obtain from $p$ by fixing all but $m$ variables, chosen arbitrarily (e.g. $q(x_1,\ldots, x_m) = p(x_1,\ldots,x_m, 1,2,1,1,..,2)$).

Does the distribution of $q$'s is also uniform over the polynomials with $m$ variables? More specifically, does $q$ have properties of a random polynomial (e.g. expected number of monomials, degree, etc.)?

Seems to me that the answer is yes, since this operation is a projection onto a subspace of dimension $m$, and a projection of a random vector is a random vector in the projected subspace. Yet I am not sure if this argument holds for an arbitrary projection (and not just for a random projection).

Larry a.
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  • Yes. The claim is obvious for the set of all functions, and (see my answer to your previous question) here every function comes from a polynomial. Of course, if you draw polynomials from a finite set that does not intersect the cosets of the key ideal $I=\langle x_i^2-1\mid i=1,2,\ldots,n\rangle$ uniformly, then you will get non-uniformity here also. – Jyrki Lahtonen May 01 '20 at 04:36
  • Thanks! May you please explain the last claim? If I draw a polynomial uniformly at random, why does the property you mentioned hold? – Larry a. May 01 '20 at 05:57

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